In this section we give the trapezium type inequalities by using distance-disturbed \(( \ell _{1} ,h_{1})\)-\(( \ell _{2},h_{2})\)-convex functions with modulus \(\mu _{1}, \mu _{2} >0\) of higher orders \((\sigma _{1},\sigma _{2})\), where \(\sigma _{1},\sigma _{2}>0\) on the coordinates on Ω.
Theorem 3.1
Suppose that \(\rho :\Omega \rightarrow \mathbb{R}\) is a distance-disturbed \(( \ell _{1} ,h_{1})\)-\(( \ell _{2},h_{2})\)-convex function of higher orders on the coordinates on Ω and \(\rho \in L_{1}(\Omega )\). Then one has the inequalities
$$\begin{aligned} &\frac{1}{4h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) + A \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{4 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu , k_{1}, k_{2} }\rho ( \hat{b},\hat{d} ) \\ &\qquad {}+{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2}}\rho ( \hat{b},\hat{c} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a}, \hat{d} ) + {}^{{\ell _{1},\ell _{2}}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\mu \nu }{4k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a}, \hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{1}\,d \imath _{2} \\ &\qquad {}- \frac{\mu _{1} [ (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}+ (\hat{a}^{\ell _{1}} -\hat{b}^{\ell _{1}} )^{\sigma _{1}} ]+\mu _{2} [ (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}+ (\hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} )^{\sigma _{2}} ]}{2}, \end{aligned}$$
(3.1)
where
$$ A=-\frac{\mu \nu }{16k_{1}k_{2}} \biggl[\frac{\mu _{1}k_{2}}{\nu } \biggl( \frac{1}{2\hat{b}^{\ell _{1}}} \biggr)^{\frac{\mu }{k_{1}}}C_{1}+ \frac{\mu _{2}k_{1}}{\mu } \biggl( \frac{1}{2\hat{d}^{\ell _{2}}} \biggr)^{\frac{\nu }{k_{2}}}C_{2} \biggr] $$
and
$$ C_{1}= \int _{\hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}}}^{\hat{a}^{\ell _{1}}+ \hat{b}^{\ell _{1}}}\imath _{1}^{\sigma _{1}} \bigl[\imath _{1}-\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}} \bigr]^{\frac{\mu }{k_{1}}-1} \,d\imath _{1}, \qquad C_{2}= \int _{\hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}}}^{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}\imath _{2}^{\sigma _{2}} \bigl[\imath _{2}- \hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}} \bigr]^{\frac{\nu }{k_{2}}-1} \,d \imath _{2}. $$
Proof
Let \(x^{\ell _{1}}=\imath _{1}\hat{a}^{\ell _{1}}+ ( 1-\imath _{1} ) \hat{b}^{\ell _{1}}\), \(y^{\ell _{1}}= ( 1-\imath _{1} ) \hat{a}^{\ell _{1}}+ \imath _{1}\hat{b}^{\ell _{1}}\) and \(u^{\ell _{2}}=\imath _{2}\hat{c}^{\ell _{2}}+ ( 1-\imath _{2} ) \hat{d}^{\ell _{2}}\), \(v^{\ell _{2}}= ( 1-\imath _{2} ) \hat{c}^{\ell _{2}}+ \imath _{2}\hat{d}^{\ell _{2}}\), then, by coordinated distance-disturbed \(( \ell _{1} ,h_{1})\)-\(( \ell _{2},h_{2})\)-convexity of higher orders of ρ, we have
$$\begin{aligned} &\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad = \rho \biggl( \biggl[ \frac{x^{\ell _{1}}+y^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}}, \biggl[ \frac{u^{\ell _{2}}+v^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq h_{1} \biggl( \frac{1}{2} \biggr) h_{2} \biggl( \frac{1}{2} \biggr) \bigl[ \rho ( x,u ) +\rho ( x,v ) + \rho ( y,u ) +\rho ( y,v ) \bigr] \\ &\qquad {}-\frac{\mu _{1}}{4} \bigl(y^{\ell _{1}}-x^{\ell _{1}} \bigr)^{\sigma _{1}}-\frac{\mu _{2}}{4} \bigl(v^{\ell _{2}}-u^{\ell _{2}} \bigr)^{\sigma _{2}}. \end{aligned}$$
Multiplying by \(\frac{\mu \nu }{4k_{1}k_{2}}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1}\) and integrating over \(( [ 0,1 ] \times [ 0,1 ] ) \), one has
$$\begin{aligned} &\frac{1}{4h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr]^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\mu \nu }{4k_{1}k_{2}} \int _{0}^{1} \int _{0}^{1} \imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ \rho ( x,u ) +\rho ( x,v )+\rho ( y,u ) +\rho ( y,v ) \bigr] \,d\imath _{1}\,d \imath _{2} \\ &\qquad {}- \frac{\mu \nu }{16k_{1}k_{2}h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) } \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[\mu _{1} \bigl(y^{\ell _{1}}-x^{\ell _{1}} \bigr)^{\sigma _{1}}+\mu _{2} \bigl(v^{\ell _{2}}-u^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr]\,d \imath _{1}\,d\imath _{2}. \end{aligned}$$
(3.2)
Note that by the change of variable, we have for the first integral on the right-hand side of the inequality (3.2)
$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ \rho ( x,u ) +\rho ( x,v ) +\rho ( y,u ) +\rho ( y,v ) \bigr] \,d\imath _{1}\,d\imath _{2} \\ &\quad =\frac{\ell _{1}\ell _{2}}{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \biggl[ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}}{ (\hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} )^{1-\frac{\nu }{k_{2}} }}\rho ( x,y ) \,dy \,dx \\ &\qquad {}+ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}}{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{1-\frac{\nu }{k_{2}} }}\rho ( x,y ) \,dy \,dx \\ &\qquad {}+ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}}{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} )^{1-\frac{\nu }{k_{2}} }}\rho ( x,y ) \,dy \,dx \\ &\qquad {}+ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}}{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{1-\frac{\nu }{k_{2}} }}\rho ( x,y ) \,dy \,dx \biggr]. \end{aligned}$$
Now applying Definition 2.22 of the Katugampola \((k_{1}, k_{2})\)-fractional integral, the first inequality of (3.1) is obtained. For the second inequality on the right-hand side of (3.1), we use the coordinated distance-disturbed \(( \ell _{1} ,h_{1})\)-\(( \ell _{2},h_{2})\)-convexity of higher orders of ρ as follows:
$$\begin{aligned}& \begin{aligned}[b] \rho ( x,u ) &=\rho \bigl( \bigl[ \imath _{1} \hat{a}^{\ell _{1}}+ ( 1-\imath _{1} ) \hat{b}^{\ell _{1}} \bigr]^{\frac{1}{\ell _{1}}}, \bigl[ \imath _{2}\hat{c}^{\ell _{2}}+ ( 1- \imath _{2} ) \hat{d}^{\ell _{2}} \bigr] ^{\frac{1}{\ell _{2}}} \bigr) \\ &\leq h_{1} ( \imath _{1} ) h_{2} ( \imath _{2} ) \rho ( \hat{a},\hat{c} ) +h_{1} ( \imath _{1} ) h_{2} ( 1-\imath _{2} ) \rho ( \hat{a}, \hat{d} ) +h_{1} ( 1-\imath _{1} ) h_{2} ( \imath _{2} ) \rho ( \hat{b},\hat{c} ) \\ &\quad {}+h_{1} ( 1-\imath _{1} ) h_{2} ( 1-\imath _{2} ) \rho ( \hat{b},\hat{d} )-\mu _{1} \bigl(\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}} -\mu _{2} \bigl( \hat{d}^{\ell _{2}}- \hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}} , \end{aligned} \end{aligned}$$
(3.3)
$$\begin{aligned}& \begin{aligned}[b] \rho ( x,v ) &=\rho \bigl( \bigl[ \imath _{1}\hat{a}^{\ell _{1}}+ ( 1-\imath _{1} ) \hat{b}^{\ell _{1}} \bigr] ^{\frac{1}{\ell _{1}}}, \bigl[ ( 1-\imath _{2} ) \hat{c}^{\ell _{2}}+\imath _{2}\hat{d}^{\ell _{2}} \bigr]^{\frac{1}{\ell _{2}}} \bigr) \\ &\leq h_{1} ( \imath _{1} ) h_{2} ( 1-\imath _{2} ) \rho ( \hat{a},\hat{c} ) +h_{1} ( \imath _{1} ) h_{2} ( \imath _{2} ) \rho ( \hat{a}, \hat{d} ) +h_{1} ( 1-\imath _{1} ) h_{2} ( 1- \imath _{2} ) \rho ( \hat{b},\hat{c} ) \\ &\quad {}+h_{1} ( 1-\imath _{1} ) h_{2} ( \imath _{2} ) \rho ( \hat{b},\hat{d} )-\mu _{1} \bigl( \hat{b}^{\ell _{1}}- \hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}-\mu _{2} \bigl(\hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} , \end{aligned} \end{aligned}$$
(3.4)
$$\begin{aligned}& \begin{aligned}[b] \rho ( y,u ) &=\rho \bigl( \bigl[ ( 1-\imath _{1} ) \hat{a}^{\ell _{1}}+\imath _{1}\hat{b}^{\ell _{1}} \bigr] ^{\frac{1}{\ell _{1}}}, \bigl[ \imath _{2}\hat{c}^{\ell _{2}}+ ( 1-\imath _{2} ) \hat{d}^{\ell _{2}} \bigr] ^{\frac{1}{\ell _{2}}} \bigr) \\ &\leq h_{1} ( 1-\imath _{1} ) h_{2} ( \imath _{2} ) \rho ( \hat{a},\hat{c} ) +h_{1} ( 1-\imath _{1} ) h_{2} ( 1-\imath _{2} ) \rho ( \hat{a}, \hat{d} ) +h_{1} ( \imath _{1} ) h_{2} ( \imath _{2} ) \rho ( \hat{b},\hat{c} ) \\ &\quad {}+h_{1} ( \imath _{1} ) h_{2} ( 1-\imath _{2} ) \rho ( \hat{b},\hat{d} )-\mu _{1} \bigl( \hat{a}^{\ell _{1}}- \hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}}-\mu _{2} \bigl(\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}} ,\end{aligned} \end{aligned}$$
(3.5)
and
$$\begin{aligned} \rho ( y,v ) =&\rho \bigl( \bigl[ ( 1-\imath _{1} ) \hat{a}^{\ell _{1}}+\imath _{1} \hat{b}^{\ell _{1}} \bigr] ^{\frac{1}{\ell _{1}}}, \bigl[ ( 1-\imath _{2} ) \hat{c}^{\ell _{2}}+\imath _{2}\hat{d}^{\ell _{2}} \bigr]^{\frac{1}{\ell _{2}}} \bigr) \\ \leq &h_{1} ( 1-\imath _{1} ) h_{2} ( 1-\imath _{2} ) \rho ( \hat{a},\hat{c} ) +h_{1} ( 1-\imath _{1} ) h_{2} ( \imath _{2} ) \rho ( \hat{a}, \hat{d} ) +h_{1} ( \imath _{1} ) h_{2} ( 1- \imath _{2} ) \rho ( \hat{b},\hat{c} ) \\ &{} +h_{1} ( \imath _{1} ) h_{2} ( \imath _{2} ) \rho ( \hat{b},\hat{d} )-\mu _{1} \bigl( \hat{a}^{\ell _{1}}- \hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}}-\mu _{2} \bigl(\hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} . \end{aligned}$$
(3.6)
Adding inequalities (3.3), (3.4), (3.5) and (3.6), we arrive at the result
$$\begin{aligned} &\rho ( x,u ) +\rho ( x,v ) +\rho ( y,u ) +\rho ( y,v ) \\ &\quad \leq \bigl[ \rho ( \hat{a},\hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \bigl[ h_{1} ( \imath _{1} ) h_{2} ( \imath _{2} ) +h_{1} ( \imath _{1} ) h_{2} ( 1- \imath _{2} ) \\ &\qquad {}+ h_{1} ( 1-\imath _{1} ) h_{2} ( \imath _{2} ) +h_{1} ( 1-\imath _{1} ) h_{2} ( 1-\imath _{2} ) \bigr] \\ &\qquad {}-2\mu _{1} \bigl[ \bigl(\hat{b}^{\ell _{1}}- \hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl(\hat{a}^{\ell _{1}}- \hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] \\ &\qquad {}-2\mu _{2} \bigl[ \bigl(\hat{d}^{\ell _{2}}- \hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl(\hat{c}^{\ell _{2}}- \hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr]. \end{aligned}$$
(3.7)
Multiplying (3.7) by \(\frac{\mu \nu }{4k_{1}k_{2}}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1}\) and integrating over \(( [0,1]\times {}[ 0,1] ) \), one has the second inequality of (3.1) by applying Definition 2.22, which then completes the proof. □
Corollary 3.2
Taking \(k_{1}, k_{2}\to 1\) in Theorem 3.1, we have
$$\begin{aligned} &\frac{1}{4h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} )} \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) + A^{\star } \\ &\quad \leq \frac{\ell _{1}^{\mu }\ell _{2}^{\nu }\Gamma ( \mu +1 ) \Gamma ( \nu +1 ) }{4 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\mu } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu }\rho ( \hat{b},\hat{d} ) \\ &\qquad {}+{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu } \rho ( \hat{b},\hat{c} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu }\rho ( \hat{a},\hat{d} ) + {}^{{\ell _{1},\ell _{2}}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu }\rho ( \hat{a}, \hat{c} ) \bigr] \\ &\quad \leq \frac{\mu \nu }{4} \bigl[ \rho ( \hat{a},\hat{c} ) + \rho ( \hat{a}, \hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\mu -1} \imath _{2}^{\nu -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{1}\,d \imath _{2} \\ &\qquad {}- \frac{\mu _{1} [ (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}+ (\hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} )^{\sigma _{1}} ]+\mu _{2} [ (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}+ (\hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} )^{\sigma _{2}} ]}{2}, \end{aligned}$$
(3.8)
where
$$ A^{\star }=-\frac{\mu \nu }{16} \biggl[\frac{\mu _{1}}{\nu } \biggl( \frac{1}{2\hat{b}^{\ell _{1}}} \biggr)^{\mu }C_{1}^{\star }+ \frac{\mu _{2}}{\mu } \biggl(\frac{1}{2\hat{d}^{\ell _{2}}} \biggr)^{\nu }C_{2}^{\star } \biggr] $$
and
$$ C_{1}^{\star }= \int _{\hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}}}^{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}\imath _{1}^{\sigma _{1}} \bigl[\imath _{1}-\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}} \bigr]^{\mu -1} \,d\imath _{1},\qquad C_{2}^{\star }= \int _{\hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}}}^{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}\imath _{2}^{\sigma _{2}} \bigl[\imath _{2}-\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}} \bigr]^{\nu -1} \,d\imath _{2}. $$
Corollary 3.3
Taking \(\mu _{1}, \mu _{2}\to 0^{+}\) in Theorem 3.1, we get
$$\begin{aligned} &\frac{1}{4h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} )} \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{4 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu , k_{1}, k_{2} }\rho ( \hat{b},\hat{d} ) \\ &\qquad {}+{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{b},\hat{c} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a}, \hat{d} ) + {}^{{\ell _{1},\ell _{2}}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\mu \nu }{4k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1- \imath _{2} ) \bigr] \,d\imath _{1}\,d\imath _{2}. \end{aligned}$$
(3.9)
Corollary 3.4
Taking \(k_{1}, k_{2}\to 1\) in Corollary 3.3, we obtain
$$\begin{aligned} &\frac{1}{4h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr]^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\ell _{1}^{\mu }\ell _{2}^{\nu }\Gamma ( \mu +1 ) \Gamma ( \nu +1 ) }{4 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu }\rho ( \hat{b},\hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu }\rho ( \hat{b},\hat{c} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu }\rho ( \hat{a}, \hat{d} ) + {}^{{\ell _{1},\ell _{2}}}I_{\hat{b}-, \hat{d}-}^{\mu ,\nu }\rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\mu \nu }{4} \bigl[ \rho ( \hat{a},\hat{c} ) + \rho ( \hat{a}, \hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\mu -1} \imath _{2}^{\nu -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{1}\,d \imath _{2}. \end{aligned}$$
(3.10)
Remark 3.5
If \(\mu =1=\nu \), then Corollary 3.4 becomes Theorem 2.16 which was proved in [38].
Remark 3.6
If \(h_{1}(t)=t=h_{2}(t)\), then Remark 3.5 coincides with Theorem 2.11 which was proved in [27].
Corollary 3.7
Taking \(\ell _{1}=\ell _{2}=1\) in Corollary 3.4, we have
$$\begin{aligned} &\frac{1}{4h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \frac{\hat{a}+b}{2}, \frac{\hat{c}+\hat{d}}{2} \biggr) \\ &\quad \leq \frac{\Gamma ( \mu +1 ) \Gamma ( \nu +1 ) }{4 ( \hat{b}-\hat{a} ) ^{\mu } ( \hat{d}-\hat{c} ) ^{\nu }} \bigl[ I_{\hat{a}+,\hat{c}+}^{\mu ,\nu } \rho ( \hat{b},\hat{d} ) + I_{\hat{a}+,\hat{d}-}^{\mu ,\nu }\rho ( \hat{b},\hat{c} ) + I_{\hat{b}-,\hat{c}+}^{\mu ,\nu }\rho ( \hat{a},\hat{d} ) + I_{\hat{b}-,\hat{d}-}^{\mu ,\nu } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\mu \nu }{4} \bigl[ \rho ( \hat{a},\hat{c} ) + \rho ( \hat{a}, \hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\mu -1} \imath _{2}^{\nu -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1- \imath _{2} ) \bigr] \,d\imath _{1}\,d\imath _{2}. \end{aligned}$$
This result coincides with Theorem 2.1 of [36], if \(h_{1}(t)=h_{2}(t)=h(t)\). Furthermore, if \(\mu =\nu =1\), it reduces to Theorem 7 of [23].
Remark 3.8
If \(h_{1}(t)=h_{2}(t)=t\) then Corollary 3.7 coincides with Theorem 2.8.
Corollary 3.9
Suppose that \(\rho :\Omega \rightarrow \mathbb{R}\) is distance-disturbed \(( \ell _{1} ,s_{1})\)-\(( \ell _{2},s_{2})\)-convex function of higher orders on the coordinates on Ω and \(\rho \in L_{1}(\Omega )\). Then one has the inequalities
$$\begin{aligned} &2^{s_{1}+s_{2}-2}\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) + A \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{4 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{d} ) + {}^{\ell _{1}, \ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{b},\hat{c} ) \\ &\qquad {}+ {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a},\hat{d} ) + {}^{{\ell _{1},\ell _{2}}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\mu \nu }{4k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{1}{(\mu +s_{1})(\nu +s_{2})}+ \frac{B(\nu ,s_{2}+1)}{(\mu +s_{1})} \\ &\qquad {}+\frac{B(\mu ,s_{1}+1)}{(\nu +s_{2})} + B(\nu ,s_{2}+1)B( \mu ,s_{1}+1) \biggr\} \\ &\qquad {}- \frac{\mu _{1} [ (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}+ (\hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} )^{\sigma _{1}} ]+\mu _{2} [ (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}+ (\hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} )^{\sigma _{2}} ]}{2}, \end{aligned}$$
where \(B(x,y)=\int _{0}^{1}\jmath ^{x-1}(1-\jmath )^{y-1}\,d\jmath \), for all \(x,y>0\), is the Beta function.
Corollary 3.10
Suppose that \(\rho :\Omega \rightarrow \mathbb{R} \) is distance-disturbed \(( \ell ,s_{1})\)-\(( \ell ,s_{2})\)-convex function of higher orders on the coordinates on Ω and \(\rho \in L_{1}(\Omega )\). Then one has the inequalities
$$\begin{aligned} &2^{s_{1}+s_{2}-2}\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr]^{\frac{1}{\ell }}, \biggl[ \frac{\hat{c}^{\ell }+\hat{d}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }} \biggr) + B \\ &\quad \leq \frac{\ell ^{\frac{\mu }{k_{1}} + \frac{\nu }{k_{2}}}\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{4 ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[^{\ell ,\ell }I_{\hat{a}+,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{d} ) + {}^{\ell ,\ell }I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{c} ) \\ &\qquad {}+{}^{\ell ,\ell }I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a},\hat{d} ) + {}^{{\ell ,\ell }}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\mu \nu }{4k_{1}k_{2}} \bigl[ \rho ( \hat{a},\hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{1}{(\mu +s_{1})(\nu +s_{2})}+ \frac{B(\nu ,s_{2}+1)}{(\mu +s_{1})}+ \frac{B(\mu ,s_{1}+1)}{(\nu +s_{2})} + B(\nu ,s_{2}+1)B(\mu ,s_{1}+1) \biggr\} \\ &\qquad {}- \frac{\mu _{1} [ (\hat{b}^{\ell }-\hat{a}^{\ell } )^{\sigma _{1}}+ (\hat{a}^{\ell }-\hat{b}^{\ell } )^{\sigma _{1}} ]+\mu _{2} [ (\hat{d}^{\ell }-\hat{c}^{\ell } )^{\sigma _{2}}+ (\hat{c}^{\ell }-\hat{d}^{\ell } )^{\sigma _{2}} ]}{2}, \end{aligned}$$
where
$$ B=-\frac{\mu \nu }{16k_{1}k_{2}} \biggl[\frac{\mu _{1}k_{2}}{\nu } \biggl( \frac{1}{2\hat{b}^{\ell }} \biggr)^{\frac{\mu }{k_{1}}}B_{1}+ \frac{\mu _{2}k_{1}}{\mu } \biggl( \frac{1}{2\hat{d}^{\ell }} \biggr)^{\frac{\nu }{k_{2}}}B_{2} \biggr] $$
and
$$ B_{1}= \int _{\hat{a}^{\ell }-\hat{b}^{\ell }}^{\hat{a}^{\ell }+\hat{b}^{\ell }}\imath _{1}^{\sigma _{1}} \bigl[\imath _{1}-\hat{a}^{\ell }+ \hat{b}^{\ell } \bigr]^{\frac{\mu }{k_{1}}-1} \,d\imath _{1},\qquad B_{2}= \int _{\hat{c}^{\ell }-\hat{d}^{\ell }}^{\hat{c}^{\ell }+\hat{d}^{\ell }} \imath _{2}^{\sigma _{2}} \bigl[\imath _{2}-\hat{c}^{\ell }+\hat{d}^{\ell } \bigr]^{\frac{\nu }{k_{2}}-1} \,d\imath _{2}. $$
Corollary 3.11
Suppose that \(\rho :\Omega \rightarrow \mathbb{R}\) is distance-disturbed \(( s_{1},s_{2} ) \)-convex function of order 2 on the coordinates on Ω and \(\rho \in L_{1}(\Omega )\). Then one has the inequalities:
$$\begin{aligned} &2^{s_{1}+s_{2}-2}\rho \biggl( \frac{\hat{a}+\hat{b}}{2}, \frac{\hat{c}+\hat{d}}{2} \biggr) + D \\ &\quad \leq \frac{\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{4 ( \hat{b}-\hat{a} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}-\hat{c} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[ {}^{1,1}I_{\hat{a}+,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{d} ) + {}^{1,1}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{c} )\\ &\qquad {} + {}^{1,1}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a},\hat{d} ) + {}^{{1,1}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\mu \nu }{4k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{1}{(\mu +s_{1})(\nu +s_{2})}+ \frac{B(\nu ,s_{2}+1)}{(\mu +s_{1})}+ \frac{B(\mu ,s_{1}+1)}{(\nu +s_{2})} + B(\nu ,s_{2}+1)B(\mu ,s_{1}+1) \biggr\} \\ &\qquad {} - \bigl[\mu _{1} (\hat{b}-\hat{a} )^{2}+\mu _{2} ( \hat{d}-\hat{c} )^{2} \bigr], \end{aligned}$$
where
$$\begin{aligned}& D=-\frac{\mu \nu }{16k_{1}k_{2}} \biggl[\frac{\mu _{1}k_{2}}{\nu } \biggl( \frac{1}{2\hat{b}} \biggr)^{\frac{\mu }{k_{1}}}D_{1}+ \frac{\mu _{2}k_{1}}{\mu } \biggl( \frac{1}{2\hat{d}} \biggr)^{\frac{\nu }{k_{2}}}D_{2} \biggr], \\& D_{1}= \int _{\hat{a}-\hat{b}}^{\hat{a}+\hat{b}}\imath _{1}^{2} [ \imath _{1}-\hat{a}+\hat{b} ]^{\frac{\mu }{k_{1}}-1} \,d\imath _{1}, \qquad D_{2}= \int _{\hat{c}-\hat{d}}^{\hat{c}+\hat{d}}\imath _{2}^{2} [\imath _{2}-\hat{c}+\hat{d} ]^{\frac{\nu }{k_{2}}-1} \,d\imath _{2}. \end{aligned}$$
To prove our next result, we need Proposition 3.12.
Proposition 3.12
Let \(\rho :I=[\hat{a},\hat{b}]\subseteq ( 0,\infty ) \rightarrow \mathbb{R} \) be a distance-disturbed \(( \ell ,h ) \)-convex function of higher order \(\sigma >0\) and \(\rho \in L_{1}[\hat{a},\hat{b}]\). Then, for \(\alpha , \mu , k>0\), the following double inequality holds:
$$\begin{aligned} &\frac{1}{h ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }} \biggr) +\frac{\alpha \mu }{4k} \bigl( \hat{b}^{\ell }-\hat{a}^{\ell } \bigr)^{\sigma }W \\ &\quad \leq \frac{\ell ^{\frac{\mu }{k} }\Gamma _{k} ( \alpha +k ) }{ ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k} }} \bigl[ ^{\ell }I_{\hat{a}+}^{\alpha , k } \rho ( \hat{b}) + {}^{\ell }I_{\hat{b}-}^{\alpha , k }\rho ( \hat{a} ) \bigr] \\ &\quad \leq \alpha \biggl[ \frac{\rho ( \hat{a} ) +\rho ( \hat{b} )}{k} \biggr] \int _{0}^{1}t^{\frac{\alpha }{k} -1} \bigl[ h ( t ) +h ( 1-t ) \bigr] \,dt \\ &\qquad {}-\mu \frac{\alpha k}{(\alpha +k)(\alpha +2k)} \bigl[ \bigl(\hat{b}^{\ell }- \hat{a}^{\ell } \bigr)^{\sigma }+ \bigl(\hat{a}^{\ell }- \hat{b}^{\ell } \bigr)^{\sigma } \bigr], \end{aligned}$$
(3.11)
where
$$ W= \int _{0}^{1}t^{\frac{\alpha }{k} -1}(2t-1)^{\sigma } \,dt. $$
Proof
Since ρ is a distance-disturbed \(( \ell ,h ) \)-convex function of higher order \(\sigma >0\) on \([\hat{a},\hat{b}]\), taking \(x^{\ell }=t\hat{a}^{\ell }+ ( 1-t ) \hat{b}^{\ell }\) and \(y^{\ell }= ( 1-t ) \hat{a}^{\ell }+t\hat{b}^{\ell }\), for all \(t\in {}[ 0,1]\), we have
$$\begin{aligned} \frac{1}{h ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }} \biggr)&\leq \rho \bigl( \bigl[t\hat{a}^{\ell }+(1-t) \hat{b}^{\ell }\bigr]^{\frac{1}{\ell }} \bigr) +\rho \bigl( \bigl[(1-t) \hat{a}^{\ell }+t\hat{b}^{\ell }\bigr]^{\frac{1}{\ell }} \bigr) \\ &\quad {}-\frac{\mu }{4}(2t-1)^{\sigma } \bigl(\hat{b}^{\ell }- \hat{a}^{\ell } \bigr)^{\sigma }. \end{aligned}$$
(3.12)
Multiplying both sides of (3.12) by \(t^{\frac{\alpha }{k} -1}\) and integrating w.r.t. t over \([0,1]\), we obtain
$$\begin{aligned} &\frac{k}{\alpha h ( \frac{1}{2} ) }f \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr]^{\frac{1}{\ell }} \biggr) \\ &\quad \leq \int _{0}^{1}t^{\frac{\alpha }{k} -1}\rho \bigl( \bigl[ t \hat{a}^{\ell }+ ( 1-t ) \hat{b}^{\ell } \bigr] ^{\frac{1}{\ell }} \bigr)\,dt \\ &\qquad {}+ \int _{0}^{1}t^{\frac{\alpha }{k} -1}\rho \bigl( \bigl[ ( 1-t ) \hat{a}^{\ell }+t\hat{b}^{\ell } \bigr] ^{\frac{1}{\ell }} \bigr) \,dt-\frac{\mu }{4} \bigl(\hat{b}^{\ell }-\hat{a}^{\ell } \bigr)^{\sigma } \int _{0}^{1}t^{\frac{\alpha }{k} -1}(2t-1)^{\sigma } \,dt. \end{aligned}$$
(3.13)
By a change of variable in (3.13), we get
$$\begin{aligned} \frac{k}{\alpha h ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr]^{\frac{1}{\ell }} \biggr) &\leq \frac{\ell }{ ( \hat{b}^{\ell }-\hat{a}^{\ell } )^{\frac{\alpha }{k} }} \biggl[ \int _{\hat{a}}^{\hat{b}} \frac{x^{\frac{\alpha }{k}-1}}{ ( \hat{b}^{\ell }-x^{\ell } ) ^{\frac{\alpha }{k} }}\rho ( x ) \,dx+ \int _{\hat{a}}^{\hat{b}} \frac{x^{\frac{\alpha }{k}-1}}{ ( x^{\ell }-\hat{a}^{\ell } ) ^{\frac{\alpha }{k}}} \rho ( x ) \,dx \biggr] \\ &\quad {}-\frac{\mu }{4} \bigl(\hat{b}^{\ell }-\hat{a}^{\ell } \bigr)^{\sigma }W. \end{aligned}$$
Applying Definition 2.20 of Katugampola k-fractional integrals, one has the first inequality of (3.11). For the second inequality on the right-hand side of (3.11), by using the distance-disturbed \(( \ell ,h ) \)-convexity of higher order \(\sigma >0\) of ρ, we have
$$\begin{aligned} \rho \bigl( \bigl[ t\hat{a}^{\ell }+ ( 1-t ) \hat{b}^{\ell } \bigr] ^{\frac{1}{\ell }} \bigr) +\rho \bigl( \bigl[ ( 1-t ) \hat{a}^{\ell }+t \hat{b}^{\ell } \bigr] ^{\frac{1}{\ell }} \bigr) &\leq \bigl[ \rho ( \hat{a}) +\rho ( \hat{b}) \bigr] \bigl( h ( t ) +h ( 1-t ) \bigr) \\ &\quad {}-\mu t(1-t) \bigl[ \bigl(\hat{b}^{\ell }-\hat{a}^{\ell } \bigr)^{\sigma }+ \bigl(\hat{a}^{\ell }-\hat{b}^{\ell } \bigr)^{\sigma } \bigr]. \end{aligned}$$
Multiplying by \(t^{\frac{\alpha }{k} -1}\) on both sides and integrating over \([ 0,1 ]\), we obtained the second inequality of (3.11). □
Corollary 3.13
Taking \(\mu \to 0^{+}\) in Proposition 3.12, we have
$$\begin{aligned} \frac{1}{h ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }} \biggr) &\leq \frac{\ell ^{\frac{\alpha }{k} }\Gamma _{k} ( \alpha +k ) }{ ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\alpha }{k} }} \bigl[^{\ell }I_{\hat{a}+}^{\alpha , k }\rho ( \hat{b} ) +{}^{\ell }I_{\hat{b}-}^{\mu , k }\rho ( \hat{a} ) \bigr] \\ &\leq \alpha \biggl[ \frac{\rho ( \hat{a} ) +\rho ( \hat{b})}{k} \biggr] \int _{0}^{1}t^{\frac{\alpha }{k} -1} \bigl[ h ( t ) +h ( 1-t ) \bigr] \,dt. \end{aligned}$$
(3.14)
Remark 3.14
If \(\alpha =k=1\), then Corollary 3.13 coincides with Theorem 5 of [8].
Remark 3.15
If \(\ell =k=1\) and \(h(t)=t\), then Corollary 3.13 coincides with Theorem 2 of [35].
Now, using Proposition 3.12 we can give the following result.
Theorem 3.16
Suppose that \(\rho :\Omega \rightarrow \mathbb{R}\) is a distance-disturbed \(( \ell _{1} ,h_{1})\)-\(( \ell _{2},h_{2})\)-convex function of higher orders on the coordinates on Ω and \(\rho \in L_{1}(\Omega )\). Then one has the inequalities
$$\begin{aligned} &\frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2h_{2} ( \frac{1}{2} ) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \biggl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho \biggl( \hat{b}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr]^{\frac{1}{\ell _{2}}} \biggr) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} }\rho \biggl( \hat{a}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \biggr] \\ &\qquad {}+ \frac{\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2h_{1} ( \frac{1}{2} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \biggl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},d \biggr) +{}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},c \biggr) \biggr] \\ &\qquad {}+ \frac{\mu \nu \ell _{1}\mu _{2} (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}}{8k_{1}k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }}W_{2} (F_{1}+F_{2} )+ \frac{\mu \nu \ell _{2}\mu _{1} (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}}{8k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }}W_{1} (F_{3}+F_{4} ) \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu , k_{1}, k_{2} }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b}, \hat{c} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a}, \hat{d} ) \\ &\qquad {} +{}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a}, \hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a}, \hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}}-1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (F_{1}+F_{2} ) \\ &\qquad {}+ \frac{\mu \ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2k_{1} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho ( \hat{a},\hat{d} ) + {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho ( \hat{b},\hat{d} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho ( \hat{a}, \hat{c} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (F_{3}+F_{4} ) \\ &\quad \leq \frac{\mu \nu }{k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1- \imath _{1} ) \bigr] \,d\imath _{2}\,d\imath _{1} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (F_{1}+F_{2} ) \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (F_{3}+F_{4} ), \end{aligned}$$
(3.15)
where
$$\begin{aligned}& F_{1}= \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \,dx,\qquad F_{2}= \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \,dx, \\& F_{3}= \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \,dy,\qquad F_{4}= \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \,dy, \end{aligned}$$
and
$$ W_{1}= \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1}(2 \imath _{1}-1)^{\sigma _{1}}\,d\imath _{1},\qquad W_{2}= \int _{0}^{1} \imath _{2}^{\frac{\nu }{k_{2}} -1}(2 \imath _{2}-1)^{\sigma _{2}}\,d \imath _{2}. $$
Proof
Since \(\rho :\Omega \rightarrow \mathbb{R}\) is a distance-disturbed \(( \ell _{1},h_{1} ) \)-\(( \ell _{2},h_{2} ) \)-convex function of higher orders \((\sigma _{1}, \sigma _{2})\), then partial mapping \(\rho _{x}: [ \hat{c},\hat{d} ] \rightarrow \mathbb{R}\) defined by \(\rho _{x} ( v ) =\rho ( x,v ) \) for all \(x\in [ \hat{a},\hat{b} ] \) is distance-disturbed \(( \ell _{2},h_{2} ) \)-convex of order \(\sigma _{1}\) on \([ \hat{c},\hat{d} ] \). Similarly, \(\rho _{y}: [ \hat{a},\hat{b} ] \rightarrow \mathbb{R}\) defined by \(\rho _{y} ( u ) =\rho ( u,y ) \) for all \(y\in [ \hat{c},\hat{d} ] \) is distance-disturbed \(( \ell _{1},h_{1} ) \)-convex of order \(\sigma _{2}\) on \([ \hat{a},\hat{b} ] \). Then, by Proposition 3.12 and applying the distance-disturbed \(( \ell _{2},h_{2} ) \)-convexity of \(\rho _{x}\), we have
$$\begin{aligned} & \frac{1}{h_{2} ( \frac{1}{2} ) }\rho _{x} \biggl( \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) +\frac{\nu \mu _{2}}{4k_{2}} \bigl(\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}W_{2} \\ &\quad \leq \frac{\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho _{x}( \hat{d}) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} } \rho _{x}( \hat{c}) \bigr] \\ &\quad \leq \nu \biggl[ \frac{\rho _{x}( \hat{c}) +\rho _{x}( \hat{d})}{k_{2}} \biggr] \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d \imath _{2} \\ &\qquad {} -\mu _{2}\frac{\nu k_{2}}{(\nu +k_{2})(\nu +2k_{2})} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr], \\ & \frac{1}{h_{2} ( \frac{1}{2} ) }\rho \biggl(x, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) +\frac{\nu \mu _{2}}{4k_{2}} \bigl(\hat{d}^{\ell _{2}}- \hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}W_{2} \\ &\quad \leq \frac{\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho ( x,\hat{d} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho ( x, \hat{c} ) \bigr] . \end{aligned}$$
Or
$$\begin{aligned} &\leq \nu \biggl[ \frac{\rho ( x,\hat{c} ) +\rho ( x,\hat{d} )}{k_{2}} \biggr] \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\quad {}-\mu _{2}\frac{\nu k_{2}}{(\nu +k_{2})(\nu +2k_{2})} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr]. \end{aligned}$$
(3.16)
Integrating inequality (3.16) w.r.t. x over \([\hat{a},\hat{b}]\) after multiplying by
$$\frac{\mu \ell _{1}x^{\ell _{1}-1}}{2k_{1} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \quad \mbox{and}\quad \frac{\mu \ell _{1}x^{\ell _{1}-1}}{2k_{1} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }}, $$
respectively, we obtain
$$\begin{aligned} &\frac{\mu \ell _{1}}{2k_{1}h_{2} ( \frac{1}{2} ) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \rho \biggl( x, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \,dx \\ &\qquad {}+ \frac{\mu \nu \ell _{1}\mu _{2} (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}}{8k_{1}k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }}W_{2} \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \,dx \\ &\quad \leq \frac{\mu \nu \ell _{1}\ell _{2}}{2k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\frac{\nu }{k_{2}} } ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \\ &\qquad {}\times \biggl[ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) }{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \,dx \\ &\qquad {}+ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) }{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \,dx \biggr] \\ &\quad \leq \frac{\mu \nu \ell _{1}}{2k_{1}k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \biggl[ \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}\rho ( x,\hat{c}) }{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }}\,dx+ \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}\rho ( x,\hat{d} ) }{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }}\,dx \biggr] \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] \\ &\qquad {}\times \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \,dx \end{aligned}$$
(3.17)
and
$$\begin{aligned} &\frac{\mu \ell _{1}}{2k_{1}h_{2} ( \frac{1}{2} ) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \rho \biggl( x, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \,dx \\ &\qquad {}+ \frac{\mu \nu \ell _{1}\mu _{2} (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}}{8k_{1}k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }}W_{2} \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \,dx \\ &\quad \leq \frac{\mu \nu \ell _{1}\ell _{2}}{2k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\frac{\nu }{k_{2}} } ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \\ &\qquad {}\times \biggl[ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) }{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \,dx \\ &\qquad {} + \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) }{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \,dx \biggr] \\ &\quad \leq \frac{\mu \nu \ell _{1}}{2k_{1}k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \biggl[ \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}\rho ( x,\hat{c} ) }{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }}\,dx+ \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}\rho ( x,\hat{d} ) }{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }}\,dx \biggr] \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] \\ &\qquad {}\times \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} }} \,dx . \end{aligned}$$
(3.18)
Now again by Proposition 3.12 and applying the distance-disturbed \(( \ell _{1},h_{1} ) \)-convexity of \(\rho _{y}\), we have
$$\begin{aligned} & \frac{1}{h_{1} ( \frac{1}{2} ) }\rho _{y} \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}} \biggr)+\frac{\mu \mu _{1}}{4k_{1}} \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}W_{1} \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho _{y}( \hat{b}) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho _{y}( \hat{a}) \bigr] \\ &\quad \leq \mu \biggl[ \frac{\rho _{y} ( \hat{a} ) +\rho _{y}( \hat{b})}{k_{1}} \biggr] \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}-\mu _{1}\frac{\mu k_{1}}{(\mu +k_{1})(\mu +2k_{1})} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}} + \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr]. \end{aligned}$$
Or
$$\begin{aligned} & \frac{1}{h_{1} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2},y \biggr] ^{\frac{1}{\ell _{1}}} \biggr)+\frac{\mu \mu _{1}}{4k_{1}} \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}W_{1} \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b},y ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} }\rho ( \hat{a},y ) \bigr] \\ &\quad \leq \mu \biggl[ \frac{\rho ( \hat{a},y ) +\rho ( \hat{b},y )}{k_{1}} \biggr] \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}-\mu _{1}\frac{\mu k_{1}}{(\mu +k_{1})(\mu +2k_{1})} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}} + \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr]. \end{aligned}$$
(3.19)
Integrating (3.19) w.r.t. y over \([\hat{c},\hat{d}]\) after multiplying by
$$\frac{\nu \ell _{2}y^{\ell _{2}-1}}{2k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} } ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \quad \mbox{and}\quad \frac{\nu \ell _{2}y^{\ell _{2}-1}}{2k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} } ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}, $$
respectively, we have
$$\begin{aligned} &\frac{\nu \ell _{2}}{2k_{2}h_{1} ( \frac{1}{2} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \int _{c}^{d} \frac{y^{\ell _{2}-1}}{ ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},y \biggr) \,dy \\ &\qquad {}+ \frac{\mu \nu \ell _{2}\mu _{1} (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}}{8k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }}W_{1} \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \,dy \\ &\quad \leq \frac{\mu \nu \ell _{1}\ell _{2}}{2k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\frac{\nu }{k_{2}} } ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \\ &\qquad {}\times \biggl[ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) }{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \,dx \\ &\qquad {} + \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) }{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \,dx \biggr] \\ &\quad \leq \frac{\mu \nu \ell _{2}}{2k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \biggl[ \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}\rho ( \hat{a},y ) }{ ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy+ \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}\rho ( \hat{b},y ) }{ ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \biggr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}} -\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] \\ &\qquad {}\times \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \,dy \end{aligned}$$
(3.20)
and
$$\begin{aligned} &\frac{\nu \ell _{2}}{2k_{2}h_{1} ( \frac{1}{2} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},y \biggr) \,dy \\ &\qquad {}+ \frac{\mu \nu \ell _{2}\mu _{1} (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}}{8k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }}W_{1} \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \,dy \\ &\quad \leq \frac{\mu \nu \ell _{1}\ell _{2}}{2k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\frac{\nu }{k_{2}} } ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \\ &\qquad {}\times \biggl[ \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) }{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \,dx \\ &\qquad {} + \int _{\hat{a}}^{\hat{b}} \int _{\hat{c}}^{\hat{d}} \frac{x^{\ell _{1}-1}y^{\ell _{2}-1}\rho ( x,y ) }{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\frac{\mu }{k_{1}} } ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \,dx \biggr] \\ &\quad \leq \frac{\mu \nu \ell _{2}}{2k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \biggl[ \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}\rho ( a,y ) }{ ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy+ \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}\rho ( \hat{b},y ) }{ ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }}\,dy \biggr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl(\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl(\hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] \\ &\qquad {}\times \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\frac{\nu }{k_{2}} }} \,dy. \end{aligned}$$
(3.21)
Adding inequalities (3.17), (3.18), (3.20), (3.21) and applying Definition 2.22, one obtains
$$\begin{aligned} &\frac{\ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2h_{2} ( \frac{1}{2} ) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \biggl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho \biggl( \hat{b}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) +{}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} }\rho \biggl( \hat{a}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \biggr] \\ &\qquad {}+ \frac{\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2h_{1} ( \frac{1}{2} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \biggl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{d} \biggr) +{}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{c} \biggr) \biggr] \\ &\qquad {}+ \frac{\mu \nu \ell _{1}\mu _{2} (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}}{8k_{1}k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }}W_{2} (F_{1}+F_{2} )+ \frac{\mu \nu \ell _{2}\mu _{1} (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}}{8k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }}W_{1} (F_{3}+F_{4} ) \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[ {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b}, \hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b}, \hat{c} ) \\ &\qquad {} + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a}, \hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a}, \hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho (\hat{a}, \hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (F_{1}+F_{2} ) \\ &\qquad {}+ \frac{\mu \ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2k_{1} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho ( \hat{a}, \hat{d} ) + {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{a}, \hat{c} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{b}, \hat{c} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (F_{3}+F_{4} ), \end{aligned}$$
which are the second and third inequalities of (3.15). For the last inequality of (3.15), applying Proposition 3.12 to the last part of the above inequality, we have
$$\begin{aligned} &\frac{\nu \ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} }\rho ( \hat{b}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a}, \hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (F_{1}+F_{2} ) \\ &\qquad {}+ \frac{\mu \ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2k_{1} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho ( \hat{a}, \hat{d} ) + {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{a}, \hat{c} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{b}, \hat{c} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (F_{3}+F_{4} ) \\ &\quad \leq \frac{\mu \nu }{k_{1}k_{2}} \bigl[ \rho ( \hat{b}, \hat{c} ) +\rho ( \hat{a}, \hat{c} ) +\rho ( \hat{b}, \hat{d} ) +\rho ( \hat{a}, \hat{d} ) \bigr] \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (F_{1}+F_{2} ) \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (F_{3}+F_{4} ). \end{aligned}$$
For the first inequality of (3.15), we again use Proposition 3.12, which then completes the proof. □
Corollary 3.17
Taking \(k_{1}, k_{2}\to 1\) in Theorem 3.16, we have
$$\begin{aligned} &\frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\ell _{1}^{\mu }\Gamma ( \mu +1 ) }{2h_{2} ( \frac{1}{2} ) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu }} \biggl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu } \rho \biggl( \hat{b}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr]^{\frac{1}{\ell _{2}}} \biggr) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu }\rho \biggl( \hat{a}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \biggr] \\ &\qquad {}+ \frac{\ell _{2}^{\nu }\Gamma ( \nu +1 ) }{2h_{1} ( \frac{1}{2} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \biggl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu } \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{d} \biggr) +{}^{\ell _{2}}I_{\hat{d}-}^{\nu }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{c} \biggr) \biggr] \\ &\qquad {}+ \frac{\mu \nu \ell _{1}\mu _{2} (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}}{8 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu }}U_{2} (G_{1}+G_{2} )+ \frac{\mu \nu \ell _{2}\mu _{1} (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}}{8 ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }}U_{1} (G_{3}+G_{4} ) \\ &\quad \leq \frac{\ell _{1}^{\mu }\ell _{2}^{\nu }\Gamma ( \mu +1 ) \Gamma ( \nu +1 ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu } \rho ( \hat{b}, \hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu } \rho ( \hat{b}, \hat{c} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-, \hat{c}+}^{\mu ,\nu }\rho ( \hat{a}, \hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu } \rho ( \hat{a}, \hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \ell _{1}^{\mu }\Gamma ( \mu +1 ) }{2 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu } \rho ( \hat{b}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{a}+}^{\mu }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu }\rho ( \hat{a}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu } \rho ( \hat{a}, \hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\nu -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d \imath _{2} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}}{2(\nu +1)(\nu +2) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (G_{1}+G_{2} ) \\ &\qquad {}+ \frac{\mu \ell _{2}^{\nu }\Gamma ( \nu +1 ) }{2 ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \bigl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu } \rho ( \hat{a}, \hat{d} ) + {}^{\ell _{2}}I_{\hat{c}+}^{\nu }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu } \rho ( \hat{a}, \hat{c} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu } \rho ( \hat{b}, \hat{c} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\mu -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d \imath _{1} \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}}{2(\mu +1)(\mu +2) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (G_{3}+G_{4} ) \\ &\quad \leq \mu \nu \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a}, \hat{d} ) +\rho ( \hat{b}, \hat{c} ) +\rho ( \hat{b}, \hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\mu -1} \imath _{2}^{\nu -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{2}\,d \imath _{1} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}}{2(\nu +1)(\nu +2) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (G_{1}+G_{2} ) \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}}{2(\mu +1)(\mu +2) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (G_{3}+G_{4} ), \end{aligned}$$
where
$$\begin{aligned}& G_{1}= \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( \hat{b}^{\ell _{1}}-x^{\ell _{1}} ) ^{1-\mu }} \,dx,\qquad G_{2}= \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell _{1}-1}}{ ( x^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{1-\mu }} \,dx, \\& G_{3}= \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( \hat{d}^{\ell _{2}}-y^{\ell _{2}} ) ^{1-\nu }} \,dy,\qquad G_{4}= \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell _{2}-1}}{ ( y^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{1-\nu }} \,dy, \end{aligned}$$
and
$$ U_{1}= \int _{0}^{1}\imath _{1}^{\mu -1}(2 \imath _{1}-1)^{\sigma _{1}}\,d\imath _{1},\qquad U_{2}= \int _{0}^{1}\imath _{2}^{\nu -1}(2 \imath _{2}-1)^{\sigma _{2}}\,d\imath _{2}. $$
Corollary 3.18
Taking \(\mu _{1}, \mu _{2}\to 0^{+}\) in Theorem 3.16, we get
$$\begin{aligned} &\frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2h_{2} ( \frac{1}{2} ) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \biggl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho \biggl( \hat{b}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr]^{\frac{1}{\ell _{2}}} \biggr) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} }\rho \biggl( \hat{a}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \biggr] \\ &\qquad {}+ \frac{\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2h_{1} ( \frac{1}{2} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \biggl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{d} \biggr) +{}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{c} \biggr) \biggr] \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu , k_{1}, k_{2} }\rho ( \hat{b},\hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{c} ) \\ &\qquad {}+ {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a}, \hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b},\hat{c} ) + {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b},\hat{d} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} }\rho ( \hat{a}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a},\hat{d} ) \bigr] \\ &\qquad \times \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}}-1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\qquad {}+ \frac{\mu \ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2k_{1} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[{}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} }\rho ( \hat{a},\hat{d} ) +{}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho ( \hat{b},\hat{d} ) \\ &\qquad {}+{}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho ( \hat{a}, \hat{c} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\quad \leq \frac{\mu \nu }{k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1- \imath _{1} ) \bigr] \,d\imath _{2}\,d\imath _{1}. \end{aligned}$$
(3.22)
Corollary 3.19
Taking \(k_{1}, k_{2}\to 1\) in Corollary 3.18, we obtain
$$\begin{aligned} &\frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{\ell _{1}^{\mu }\Gamma ( \mu +1 ) }{2h_{2} ( \frac{1}{2} ) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu }} \biggl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu } \rho \biggl( \hat{b}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr]^{\frac{1}{\ell _{2}}} \biggr) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu }\rho \biggl( \hat{a}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \biggr] \\ &\qquad {}+ \frac{\ell _{2}^{\nu }\Gamma ( \nu +1 ) }{2h_{1} ( \frac{1}{2} ) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \biggl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu } \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{d} \biggr) +{}^{\ell _{2}}I_{\hat{d}-}^{\nu }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{c} \biggr) \biggr] \\ &\quad \leq \frac{\ell _{1}^{\mu }\ell _{2}^{\nu }\Gamma ( \mu +1 ) \Gamma ( \nu +1 ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu }\rho ( \hat{b},\hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu } \rho ( \hat{b},\hat{c} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-, \hat{c}+}^{\mu ,\nu }\rho ( \hat{a},\hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \ell _{1}^{\mu }\Gamma ( \mu +1 ) }{2 ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\mu }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu } \rho ( \hat{b},\hat{c} ) + {}^{\ell _{1}}I_{\hat{a}+}^{\mu }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu }\rho ( \hat{a},\hat{c} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu } \rho ( \hat{a},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\nu -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d \imath _{2} \\ &\qquad {}+ \frac{\mu \ell _{2}^{\nu }\Gamma ( \nu +1 ) }{2 ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\nu }} \bigl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu } \rho ( \hat{a},\hat{d} ) + {}^{\ell _{2}}I_{\hat{c}+}^{\nu }\rho ( \hat{b}, \hat{d} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu } \rho ( \hat{a},\hat{c} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\mu -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d \imath _{1} \\ &\quad \leq \mu \nu \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) + \rho ( \hat{b}, \hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\mu -1} \imath _{2}^{\nu -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{2}\,d \imath _{1}. \end{aligned}$$
(3.23)
Remark 3.20
If \(\mu =1=\nu \), then Corollary 3.19 gives Theorem 2.17, which was proved in [38].
Remark 3.21
If \(\ell _{1}=\ell _{2}=1\) and \(h_{1}(t)=h_{2}(t)=t\), then Remark 3.20 reduced to Theorem 2.8, which was proved in [32].
Corollary 3.22
Let ρ be a distance-disturbed \((\ell ,h_{1})\)–\((\ell ,h_{2})\)-convex function on the coordinates on Ω and \(\rho \in L_{1}(\Omega )\), then one has following inequalities:
$$\begin{aligned} &\frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr]^{\frac{1}{\ell }}, \biggl[ \frac{\hat{c}^{\ell }+\hat{d}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }} \biggr) \\ &\quad \leq \frac{\ell ^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2h_{2} ( \frac{1}{2} ) ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }} \biggl[ ^{\ell }I_{\hat{a}+}^{\mu , k_{1} } \rho \biggl( \hat{b}, \biggl[ \frac{\hat{c}^{\ell }+\hat{d}^{\ell }}{2} \biggr]^{\frac{1}{\ell }} \biggr) + {}^{\ell }I_{\hat{b}-}^{\mu , k_{1} }\rho \biggl( \hat{a}, \biggl[ \frac{\hat{c}^{\ell }+\hat{d}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }} \biggr) \biggr] \\ &\qquad {}+ \frac{\ell ^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2h_{1} ( \frac{1}{2} ) ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \biggl[ ^{\ell }I_{\hat{c}+}^{\nu , k_{2} } \rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }},\hat{d} \biggr) + {}^{\ell }I_{\hat{d}-}^{\nu , k_{2} }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }},\hat{c} \biggr) \biggr] \\ &\qquad {}+ \frac{\mu \nu \ell \mu _{2} (\hat{d}^{\ell }-\hat{c}^{\ell } )^{\sigma _{2}}}{8k_{1}k_{2} ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }}W_{2} (H_{1}+H_{2} )+ \frac{\mu \nu \ell \mu _{1} (\hat{b}^{\ell }-\hat{a}^{\ell } )^{\sigma _{1}}}{8k_{1}k_{2} ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }}W_{1} (H_{3}+H_{4} ) \\ &\quad \leq \frac{\ell ^{\frac{\mu }{k_{1}} +\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[^{\ell ,\ell }I_{\hat{a}+,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{b},\hat{d} ) + {}^{\ell ,\ell }I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{c} ) + {}^{\ell ,\ell }I_{\hat{b}-, \hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a},\hat{d} ) + {}^{\ell ,\ell }I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \ell ^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2k_{2} ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }} \bigl[ ^{\ell }I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b},\hat{c} ) + {}^{\ell }I_{\hat{a}+}^{\mu , k_{1} }\rho ( \hat{b},\hat{d} ) + {}^{\ell }I_{\hat{b}-}^{\mu , k_{1} }\rho ( \hat{a},\hat{c} ) + {}^{\ell }I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{2}^{\frac{\nu }{k_{2}}-1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \,d\imath _{2} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl(\hat{d}^{\ell }-\hat{c}^{\ell } \bigr)^{\sigma _{2}}+ \bigl(\hat{c}^{\ell }-\hat{d}^{\ell } \bigr)^{\sigma _{2}} \bigr] (H_{1}+H_{2} ) \\ &\qquad {}+ \frac{\mu \ell ^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2k_{1} ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \bigl[ ^{\ell }I_{\hat{c}+}^{\nu , k_{2} } \rho ( \hat{a},\hat{d} ) + {}^{\ell }I_{\hat{c}+}^{\nu , k_{2} }\rho ( \hat{b},\hat{d} ) + {}^{\ell }I_{\hat{d}-}^{\nu , k_{2} }\rho ( \hat{a},\hat{c} ) + {}^{\ell }I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1-\imath _{1} ) \bigr] \,d\imath _{1} \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell }-\hat{a}^{\ell } \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell }-\hat{b}^{\ell } \bigr)^{\sigma _{1}} \bigr] (H_{3}+H_{4} ) \\ &\quad \leq \frac{\mu \nu }{k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\imath _{1}^{\frac{\mu }{k_{1}} -1} \imath _{2}^{\frac{\nu }{k_{2}} -1} \bigl[ h_{2} ( \imath _{2} ) +h_{2} ( 1-\imath _{2} ) \bigr] \bigl[ h_{1} ( \imath _{1} ) +h_{1} ( 1- \imath _{1} ) \bigr] \,d\imath _{2}\,d\imath _{1} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell }-\hat{c}^{\ell } \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell }-\hat{d}^{\ell } \bigr)^{\sigma _{2}} \bigr] (H_{1}+H_{2} ) \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell }-\hat{a}^{\ell } \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell }-\hat{b}^{\ell } \bigr)^{\sigma _{1}} \bigr] (H_{3}+H_{4} ), \end{aligned}$$
(3.24)
where
$$ H_{1}= \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell -1}}{ ( \hat{b}^{\ell }-x^{\ell } ) ^{1-\frac{\mu }{k_{1}} }} \,dx, \qquad H_{2}= \int _{\hat{a}}^{\hat{b}} \frac{x^{\ell -1}}{ ( x^{\ell }-\hat{a}^{\ell } ) ^{1-\frac{\mu }{k_{1}} }} \,dx, $$
and
$$ H_{3}= \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell -1}}{ ( \hat{d}^{\ell }-y^{\ell } ) ^{1-\frac{\nu }{k_{2}} }} \,dy, \qquad H_{4}= \int _{\hat{c}}^{\hat{d}} \frac{y^{\ell -1}}{ ( y^{\ell }-\hat{c}^{\ell } ) ^{1-\frac{\nu }{k_{2}} }} \,dy. $$
Corollary 3.23
Let \(\rho : \Omega \rightarrow \mathbb{R}\) be a distance-disturbed \((\ell _{1},s_{1})\)-\((\ell _{2},s_{2})\)-convex function on the coordinates on Ω and \(\rho \in L_{1}(\Omega )\). Then one has the inequalities
$$\begin{aligned} &2^{s_{1}+s_{2}}\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr]^{\frac{1}{\ell _{1}}}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \\ &\quad \leq \frac{2^{s_{2}-1}\ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \biggl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho \biggl( \hat{b}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr]^{\frac{1}{\ell _{2}}} \biggr) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} }\rho \biggl( \hat{a}, \biggl[ \frac{\hat{c}^{\ell _{2}}+\hat{d}^{\ell _{2}}}{2} \biggr] ^{\frac{1}{\ell _{2}}} \biggr) \biggr] \\ &\qquad {}+ \frac{2^{s_{1}-1}\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times\biggl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{d} \biggr) +{}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell _{1}}+\hat{b}^{\ell _{1}}}{2} \biggr] ^{\frac{1}{\ell _{1}}},\hat{c} \biggr) \biggr] \\ &\qquad {}+ \frac{\mu \nu \ell _{1}\mu _{2} (\hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} )^{\sigma _{2}}}{8k_{1}k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }}W_{2} (F_{1}+F_{2} )+ \frac{\mu \nu \ell _{2}\mu _{1} (\hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} )^{\sigma _{1}}}{8k_{1}k_{2} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }}W_{1} (F_{3}+F_{4} ) \\ &\quad \leq \frac{\ell _{1}^{\frac{\mu }{k_{1}} }\ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[{}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{c}+}^{\mu , \nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{c} ) \\ &\qquad {}+ {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a}, \hat{d} ) + {}^{\ell _{1},\ell _{2}}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \ell _{1}^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2k_{2} ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b},\hat{c} ) + {}^{\ell _{1}}I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b},\hat{d} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} }\rho ( \hat{a}, \hat{c} ) + {}^{\ell _{1}}I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl[ \frac{k_{2}}{\nu +k_{2}s_{2}}+B \biggl( \frac{\nu }{k_{2}} ,s_{2}+1 \biggr) \biggr] \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (F_{1}+F_{2} ) \\ &\qquad {}+ \frac{\mu \ell _{2}^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2k_{1} ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho ( \hat{a},\hat{d} ) + {}^{\ell _{2}}I_{\hat{c}+}^{\nu , k_{2} } \rho ( \hat{b},\hat{d} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} }\rho ( \hat{a}, \hat{c} ) + {}^{\ell _{2}}I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \biggl[ \frac{k_{1}}{\mu +k_{1}s_{1}}+B \biggl( \frac{\mu }{k_{1}} ,s_{1}+1 \biggr) \biggr] \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (F_{3}+F_{4} ) \\ &\quad \leq \frac{\mu \nu }{k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho (\hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{k_{1}k_{2}}{(\mu +k_{1}s_{1})(\nu +k_{2}s_{2})}+ \frac{k_{2}B (\frac{\mu }{k_{1}} ,s_{1}+1 )}{\nu +k_{2}s_{2}}+ \frac{k_{1}B (\frac{\nu }{k_{2}} ,s_{2}+1 )}{\mu +k_{1}s_{1}} \\ &\qquad {} +B \biggl(\frac{\mu }{k_{1}} ,s_{1}+1 \biggr)B \biggl(\frac{\nu }{k_{2}} ,s_{2}+1 \biggr) \biggr\} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell _{1}k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell _{2}}-\hat{d}^{\ell _{2}} \bigr)^{\sigma _{2}} \bigr] (F_{1}+F_{2} ) \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell _{2}k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell _{2}}-\hat{c}^{\ell _{2}} ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell _{1}}-\hat{a}^{\ell _{1}} \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell _{1}}-\hat{b}^{\ell _{1}} \bigr)^{\sigma _{1}} \bigr] (F_{3}+F_{4} ). \end{aligned}$$
Corollary 3.24
Taking \(\ell _{1}=\ell _{2}=\ell \) in Corollary 3.23, we have
$$\begin{aligned} &2^{s_{1}+s_{2}}\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr]^{\frac{1}{\ell }}, \biggl[ \frac{\hat{c}^{\ell }+\hat{d}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }} \biggr) \\ &\quad \leq \frac{2^{s_{2}-1}\ell ^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{ ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }} \biggl[ ^{\ell }I_{\hat{a}+}^{\mu , k_{1} } \rho \biggl( \hat{b}, \biggl[ \frac{\hat{c}^{\ell }+\hat{d}^{\ell }}{2} \biggr]^{\frac{1}{\ell }} \biggr) + {}^{\ell }I_{\hat{b}-}^{\mu , k_{1} }\rho \biggl( \hat{a}, \biggl[ \frac{\hat{c}^{\ell }+\hat{d}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }} \biggr) \biggr] \\ &\qquad {}+ \frac{2^{s_{1}-1}\ell ^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \biggl[ ^{\ell }I_{\hat{c}+}^{\nu , k_{2} } \rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }},\hat{d} \biggr) + {}^{\ell }I_{\hat{d}-}^{\nu , k_{2} }\rho \biggl( \biggl[ \frac{\hat{a}^{\ell }+\hat{b}^{\ell }}{2} \biggr] ^{\frac{1}{\ell }},\hat{c} \biggr) \biggr] \\ &\qquad {}+ \frac{\mu \nu \ell \mu _{2} (\hat{d}^{\ell }-\hat{c}^{\ell } )^{\sigma _{2}}}{8k_{1}k_{2} ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }}W_{2} (H_{1}+H_{2} )+ \frac{\mu \nu \ell \mu _{1} (\hat{b}^{\ell }-\hat{a}^{\ell } )^{\sigma _{1}}}{8k_{1}k_{2} ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }}W_{1} (H_{3}+H_{4} ) \\ &\quad \leq \frac{\ell ^{\frac{\mu }{k_{1}} +\frac{\nu }{k_{2}} }\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} } ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[^{\ell ,\ell }I_{\hat{a}+,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{d} ) + {}^{\ell ,\ell }I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{c} ) + {}^{\ell ,\ell }I_{\hat{b}-, \hat{c}+}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a},\hat{d} ) + {}^{\ell ,\ell }I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \ell ^{\frac{\mu }{k_{1}} }\Gamma _{k_{1}} ( \mu +k_{1} ) }{2k_{2} ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }} \bigl[ ^{\ell }I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b},\hat{c} ) + {}^{\ell }I_{\hat{a}+}^{\mu , k_{1} }\rho ( \hat{b},\hat{d} ) + {}^{\ell }I_{\hat{b}-}^{\mu , k_{1} }\rho ( \hat{a},\hat{c} ) + {}^{\ell }I_{\hat{b}-}^{\mu , k_{1} } \rho ( \hat{a},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl[ \frac{k_{2}}{\nu +k_{2}s_{2}}+B \biggl( \frac{\nu }{k_{2}} ,s_{2}+1 \biggr) \biggr] \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell }-\hat{c}^{\ell } \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell }-\hat{d}^{\ell } \bigr)^{\sigma _{2}} \bigr] (H_{1}+H_{2} ) \\ &\qquad {}+ \frac{\mu \ell ^{\frac{\nu }{k_{2}} }\Gamma _{k_{2}} ( \nu +k_{2} ) }{2k_{1} ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \bigl[ ^{\ell }I_{\hat{c}+}^{\nu , k_{2} } \rho (\hat{a},\hat{d} ) + {}^{\ell }I_{\hat{c}+}^{\nu , k_{2} }\rho ( \hat{b},\hat{d} ) + {}^{\ell }I_{\hat{d}-}^{\nu , k_{2} }\rho ( \hat{a},\hat{c} ) + {}^{\ell }I_{\hat{d}-}^{\nu , k_{2} } \rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \biggl[ \frac{k_{1}}{\mu +k_{1}s_{1}}+B \biggl( \frac{\mu }{k_{1}} ,s_{1}+1 \biggr) \biggr] \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell }-\hat{a}^{\ell } \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell }-\hat{b}^{\ell } \bigr)^{\sigma _{1}} \bigr] (H_{3}+H_{4} ) \\ &\quad \leq \frac{\mu \nu }{k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{k_{1}k_{2}}{(\mu +k_{1}s_{1})(\nu +k_{2}s_{2})}+ \frac{k_{2}B (\frac{\mu }{k_{1}} ,s_{1}+1 )}{\nu +k_{2}s_{2}} \\ &\qquad {}+ \frac{k_{1}B (\frac{\nu }{k_{2}},s_{2}+1 )}{\mu +k_{1}s_{1}} +B \biggl(\frac{\mu }{k_{1}} ,s_{1}+1 \biggr)B \biggl( \frac{\nu }{k_{2}} ,s_{2}+1 \biggr) \biggr\} \\ &\qquad {}-\mu _{2} \frac{\mu \nu \ell k_{2}}{2k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}^{\ell }-\hat{a}^{\ell } ) ^{\frac{\mu }{k_{1}} }} \bigl[ \bigl( \hat{d}^{\ell }-\hat{c}^{\ell } \bigr)^{\sigma _{2}}+ \bigl( \hat{c}^{\ell }-\hat{d}^{\ell } \bigr)^{\sigma _{2}} \bigr] (H_{1}+H_{2} ) \\ &\qquad {}-\mu _{1} \frac{\mu \nu \ell k_{1}}{2k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}^{\ell }-\hat{c}^{\ell } ) ^{\frac{\nu }{k_{2}} }} \bigl[ \bigl( \hat{b}^{\ell }-\hat{a}^{\ell } \bigr)^{\sigma _{1}}+ \bigl( \hat{a}^{\ell }-\hat{b}^{\ell } \bigr)^{\sigma _{1}} \bigr] (H_{3}+H_{4} ), \end{aligned}$$
where \(H_{1}\), \(H_{2}\), \(H_{3}\) and \(H_{4}\) are defined in Corollary 3.22.
Corollary 3.25
Taking \(\ell =1\) and \(\sigma _{1}=\sigma _{2}=2\) in Corollary 3.24, we get
$$\begin{aligned} &2^{s_{1}+s_{2}}\rho \biggl(\frac{\hat{a}+\hat{b}}{2}, \frac{\hat{c}+\hat{d}}{2} \biggr) \\ &\quad \leq \frac{2^{s_{2}-1}\Gamma _{k_{1}} ( \mu +k_{1} ) }{ ( \hat{b}-\hat{a} ) ^{\frac{\mu }{k_{1}} }} \biggl[ I_{\hat{a}+}^{\mu , k_{1} } \rho \biggl( \hat{b}, \frac{\hat{c}+\hat{d}}{2} \biggr) + I_{\hat{b}-}^{\mu , k_{1} } \rho \biggl( \hat{a}, \frac{\hat{c}+\hat{d}}{2} \biggr) \biggr] \\ &\qquad {}+ \frac{2^{s_{1}-1}\Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{d}-\hat{c} ) ^{\frac{\nu }{k_{2}} }} \biggl[ I_{\hat{c}+}^{\nu , k_{2} } \rho \biggl( \frac{\hat{a}+\hat{b}}{2}, \hat{d} \biggr) + I_{\hat{d}-}^{\nu , k_{2} } \rho \biggl( \frac{\hat{a}+\hat{b}}{2},\hat{c} \biggr) \biggr] \\ &\qquad {}+ \frac{\mu \nu \mu _{2} (\hat{d}-\hat{c} )^{2}}{8k_{1}k_{2} ( \hat{b}-\hat{a} ) ^{\frac{\mu }{k_{1}} }}W_{2}^{\star } \bigl(H_{1}^{\star }+H_{2}^{\star } \bigr)+ \frac{\mu \nu \mu _{1} (\hat{b}-\hat{a} )^{2}}{8k_{1}k_{2} ( \hat{d}-\hat{c} ) ^{\frac{\nu }{k_{2}} }}W_{1}^{\star } \bigl(H_{3}^{\star }+H_{4}^{\star } \bigr) \\ &\quad \leq \frac{\Gamma _{k_{1}} ( \mu +k_{1} ) \Gamma _{k_{2}} ( \nu +k_{2} ) }{ ( \hat{b}-\hat{a} ) ^{\frac{\mu }{k_{1}} } ( \hat{d}-\hat{c} ) ^{\frac{\nu }{k_{2}} }} \\ &\qquad {}\times \bigl[{}^{1,1}I_{\hat{a}+,\hat{c}+}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{d} ) + {}^{1,1}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} } \rho ( \hat{b},\hat{c} ) \\ &\qquad {}+ {}^{1,1}I_{\hat{b}-,\hat{c}+}^{\mu , \nu , k_{1}, k_{2} }\rho ( \hat{a},\hat{d} ) + {}^{1,1}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu , k_{1}, k_{2} }\rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \Gamma _{k_{1}} ( \mu +k_{1} ) }{2k_{2} ( \hat{b}-\hat{a} ) ^{\frac{\mu }{k_{1}} }} \bigl[ I_{\hat{a}+}^{\mu , k_{1} } \rho ( \hat{b},\hat{c} ) + I_{\hat{a}+}^{\mu , k_{1} }\rho ( \hat{b},\hat{d} ) + I_{\hat{b}-}^{\mu , k_{1} }\rho ( \hat{a},\hat{c} ) + I_{\hat{b}-}^{\mu , k_{1} }\rho ( \hat{a},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl[ \frac{k_{2}}{\nu +k_{2}s_{2}}+B \biggl( \frac{\nu }{k_{2}} ,s_{2}+1 \biggr) \biggr] \\ &\qquad {}-\mu _{2} \frac{\mu \nu k_{2} (\hat{d}-\hat{c} )^{2}}{k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}-\hat{a} ) ^{\frac{\mu }{k_{1}} }} \bigl(H_{1}^{\star }+H_{2}^{\star } \bigr) \\ &\qquad {}+ \frac{\mu \Gamma _{k_{2}} ( \nu +k_{2} ) }{2k_{1} ( \hat{d}-\hat{c} ) ^{\frac{\nu }{k_{2}} }} \bigl[ I_{\hat{c}+}^{\nu , k_{2} }\rho ( \hat{a},\hat{d} ) + I_{\hat{c}+}^{\nu , k_{2} }\rho ( \hat{b},\hat{d} ) + I_{\hat{d}-}^{\nu , k_{2} }\rho ( \hat{a},\hat{c} ) + I_{\hat{d}-}^{\nu , k_{2} }\rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \biggl[ \frac{k_{1}}{\mu +k_{1}s_{1}}+B \biggl( \frac{\mu }{k_{1}} ,s_{1}+1 \biggr) \biggr] \\ &\qquad {}-\mu _{1} \frac{\mu \nu k_{1} (\hat{b}-\hat{a} )^{2}}{k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}-\hat{c} ) ^{\frac{\nu }{k_{2}} }} \bigl(H_{3}^{\star }+H_{4}^{\star } \bigr) \\ &\quad \leq \frac{\mu \nu }{k_{1}k_{2}} \bigl[ \rho ( \hat{a}, \hat{c} ) +\rho ( \hat{a},\hat{d} ) +\rho ( \hat{b},\hat{c} ) +\rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{k_{1}k_{2}}{(\mu +k_{1}s_{1})(\nu +k_{2}s_{2})}+ \frac{k_{2}B (\frac{\mu }{k_{1}} ,s_{1}+1 )}{\nu +k_{2}s_{2}} \\ &\qquad {}+ \frac{k_{1}B (\frac{\nu }{k_{2}} ,s_{2}+1 )}{\mu +k_{1}s_{1}} +B \biggl(\frac{\mu }{k_{1}} ,s_{1}+1 \biggr)B \biggl(\frac{\nu }{k_{2}} ,s_{2}+1 \biggr) \biggr\} \\ &\qquad {}-\mu _{2} \frac{\mu \nu k_{2} (\hat{d}-\hat{c} )^{2}}{k_{1}(\nu +k_{2})(\nu +2k_{2}) ( \hat{b}-\hat{a} ) ^{\frac{\mu }{k_{1}} }} \bigl(H_{1}^{\star }+H_{2}^{\star } \bigr) \\ &\qquad {}-\mu _{1} \frac{\mu \nu k_{1} (\hat{b}-\hat{a} )^{2}}{k_{2}(\mu +k_{1})(\mu +2k_{1}) ( \hat{d}-\hat{c} ) ^{\frac{\nu }{k_{2}} }} \bigl(H_{3}^{\star }+H_{4}^{\star } \bigr), \end{aligned}$$
where
$$\begin{aligned}& H_{1}^{\star }= \int _{\hat{a}}^{\hat{b}} \frac{dx}{ ( \hat{b}-x ) ^{1-\frac{\mu }{k_{1}} }},\qquad H_{2}^{\star }= \int _{\hat{a}}^{\hat{b}} \frac{dx}{ ( x-\hat{a} ) ^{1-\frac{\mu }{k_{1}} }}, \\& H_{3}^{\star }= \int _{\hat{c}}^{\hat{d}} \frac{dy}{ ( \hat{d}-y ) ^{1-\frac{\nu }{k_{2}} }},\qquad H_{4}^{\star }= \int _{\hat{c}}^{\hat{d}} \frac{dy}{ ( y-\hat{c} ) ^{1-\frac{\nu }{k_{2}} }}, \end{aligned}$$
and
$$ W_{1}^{\star }=\frac{4k_{1}}{\mu +2k_{1}}-\frac{4k_{1}}{\mu +k_{1}}+ \frac{k_{1}}{\mu },\qquad W_{2}^{\star }=\frac{4k_{2}}{\nu +2k_{2}}- \frac{4k_{2}}{\nu +k_{2}}+\frac{k_{2}}{\nu }. $$
Corollary 3.26
Taking \(k_{1}, k_{2}\to 1\) in Corollary 3.25, we have
$$\begin{aligned} &2^{s_{1}+s_{2}}\rho \biggl(\frac{\hat{a}+\hat{b}}{2}, \frac{\hat{c}+\hat{d}}{2} \biggr) \\ &\quad \leq \frac{2^{s_{2}-1}\Gamma ( \mu +1 ) }{ ( \hat{b}-\hat{a} ) ^{\mu }} \biggl[ I_{\hat{a}+}^{\mu } \rho \biggl( \hat{b}, \frac{\hat{c}+\hat{d}}{2} \biggr) + I_{\hat{b}-}^{\mu } \rho \biggl( \hat{a}, \frac{\hat{c}+\hat{d}}{2} \biggr) \biggr] \\ &\qquad {}+ \frac{2^{s_{1}-1}\Gamma ( \nu +1 ) }{ ( \hat{d}-\hat{c} ) ^{\nu }} \biggl[ I_{\hat{c}+}^{\nu } \rho \biggl( \frac{\hat{a}+\hat{b}}{2},\hat{d} \biggr) + I_{\hat{d}-}^{\nu } \rho \biggl( \frac{\hat{a}+\hat{b}}{2},\hat{c} \biggr) \biggr] \\ &\qquad {}+ \frac{\mu \nu \mu _{2} (\hat{d}-\hat{c} )^{2}}{8 ( \hat{b}-\hat{a} ) ^{\mu }}W_{4}^{\star } \bigl(H_{5}^{\star }+H_{6}^{\star } \bigr)+ \frac{\mu \nu \mu _{1} (\hat{b}-\hat{a} )^{2}}{8 ( \hat{d}-\hat{c} ) ^{\nu }}W_{3}^{\star } \bigl(H_{7}^{\star }+H_{8}^{\star } \bigr) \\ &\quad \leq \frac{\Gamma ( \mu +1 ) \Gamma ( \nu +1 ) }{ ( \hat{b}-\hat{a} ) ^{\mu } ( \hat{d}-\hat{c} ) ^{\nu }} \\ &\qquad {}\times \bigl[{}^{1,1}I_{\hat{a}+,\hat{c}+}^{\mu ,\nu } \rho ( \hat{b},\hat{d} ) + {}^{1,1}I_{\hat{a}+,\hat{d}-}^{\mu ,\nu } \rho ( \hat{b},\hat{c} ) + {}^{1,1}I_{\hat{b}-,\hat{c}+}^{\mu ,\nu }\rho ( \hat{a},\hat{d} ) + {}^{1,1}I_{\hat{b}-,\hat{d}-}^{\mu ,\nu }\rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \Gamma ( \mu +1 ) }{2 ( \hat{b}-\hat{a} ) ^{\mu }} \bigl[ I_{\hat{a}+}^{\mu } \rho ( \hat{b},\hat{c} ) + I_{\hat{a}+}^{\mu }\rho ( \hat{b},\hat{d} ) + I_{\hat{b}-}^{\mu }\rho ( \hat{a},\hat{c} ) + I_{\hat{b}-}^{\mu }\rho ( \hat{a},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl[ \frac{1}{\nu +s_{2}}+B (\nu ,s_{2}+1 ) \biggr] \\ &\qquad {}-\mu _{2} \frac{\mu \nu (\hat{d}-\hat{c} )^{2}}{(\nu +1)(\nu +2) ( \hat{b}-\hat{a} ) ^{\mu }} \bigl(H_{5}^{\star }+H_{6}^{\star } \bigr) \\ &\qquad {}+ \frac{\mu \Gamma ( \nu +1 ) }{2 ( \hat{d}-\hat{c} ) ^{\nu }} \bigl[ I_{\hat{c}+}^{\nu } \rho ( \hat{a},\hat{d} ) + I_{\hat{c}+}^{\nu }\rho (\hat{b},\hat{d} ) + I_{\hat{d}-}^{\nu }\rho ( \hat{a},\hat{c} ) + I_{\hat{d}-}^{\nu }\rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \biggl[ \frac{1}{\mu +s_{1}}+B (\mu ,s_{1}+1 ) \biggr] \\ &\qquad {}-\mu _{1} \frac{\mu \nu (\hat{b}-\hat{a} )^{2}}{(\mu +1)(\mu +2) ( \hat{d}-\hat{c} ) ^{\nu }} \bigl(H_{7}^{\star }+H_{8}^{\star } \bigr) \\ &\quad \leq \mu \nu \bigl[ \rho ( \hat{a},\hat{c} ) +\rho ( \hat{a},\hat{d} ) + \rho ( \hat{b},\hat{c} ) + \rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{1}{(\mu +s_{1})(\nu +s_{2})}+ \frac{B (\mu ,s_{1}+1 )}{\nu +s_{2}}+ \frac{B (\nu ,s_{2}+1 )}{\mu +s_{1}} +B (\mu ,s_{1}+1 )B (\nu ,s_{2}+1 ) \biggr\} \\ &\qquad {}-\mu _{2} \frac{\mu \nu (\hat{d}-c )^{2}}{(\nu +1)(\nu +2) ( \hat{b}-\hat{a} ) ^{\mu }} \bigl(H_{5}^{\star }+H_{6}^{\star } \bigr)-\mu _{1} \frac{\mu \nu (\hat{b}-\hat{a} )^{2}}{(\mu +1)(\mu +2) ( \hat{d}-\hat{c} ) ^{\nu }} \bigl(H_{7}^{\star }+H_{8}^{\star } \bigr), \end{aligned}$$
where
$$\begin{aligned} &H_{5}^{\star }= \int _{\hat{a}}^{\hat{b}} \frac{dx}{ ( \hat{b}-x ) ^{1-\mu }},\qquad H_{6}^{\star }= \int _{\hat{a}}^{\hat{b}} \frac{dx}{ ( x-\hat{a} ) ^{1-\mu }}, \\ &H_{7}^{\star }= \int _{\hat{c}}^{\hat{d}} \frac{dy}{ ( \hat{d}-y ) ^{1-\nu }},\qquad H_{8}^{\star }= \int _{\hat{c}}^{\hat{d}} \frac{dy}{ ( y-\hat{c} ) ^{1-\nu }}, \end{aligned}$$
and
$$ W_{3}^{\star }=\frac{4}{\mu +2}-\frac{4}{\mu +1}+ \frac{1}{\mu },\qquad W_{4}^{\star }=\frac{4}{\nu +2}- \frac{4}{\nu +1}+\frac{1}{\nu }. $$
Corollary 3.27
Taking \(\mu _{1}, \mu _{2}\to 0^{+}\) in Corollary 3.26, we obtain
$$\begin{aligned} &2^{s_{1}+s_{2}}\rho \biggl( \frac{\hat{a}+\hat{b}}{2}, \frac{\hat{c}+\hat{d}}{2} \biggr) \\ &\quad \leq \frac{2^{s_{2}-1}\Gamma ( \mu +1 ) }{ ( \hat{b}-\hat{a} )^{\mu }} \biggl[ I_{\hat{a}+}^{\mu }\rho \biggl( \hat{b}, \frac{\hat{c}+\hat{d}}{2} \biggr) +I_{\hat{b}-}^{\mu }\rho \biggl( \hat{a},\frac{\hat{c}+\hat{d}}{2} \biggr) \biggr] \\ &\qquad {}+ \frac{2^{s_{1}-1}\Gamma ( \nu +1 ) }{ ( \hat{d}-\hat{c} ) ^{\nu }} \biggl[ I_{\hat{c}+}^{\nu }\rho \biggl( \frac{\hat{a}+\hat{b}}{2}, \hat{d} \biggr) +I_{\hat{d}-}^{\nu }\rho \biggl( \frac{\hat{a}+\hat{b}}{2},\hat{c} \biggr) \biggr] \\ &\quad \leq \frac{\Gamma ( \mu +1 ) \Gamma ( \nu +1 ) }{ ( \hat{b}-\hat{a} ) ^{\mu } ( \hat{d}-\hat{c} ) ^{\nu }} \bigl[ I_{\hat{a}+,\hat{c}+}^{\mu ,\nu } \rho ( \hat{b},\hat{d} ) + I_{\hat{a}+,\hat{d}-}^{\mu ,\nu }\rho ( \hat{b},\hat{c} ) \\ &\qquad {} + I_{\hat{b}-,\hat{c}+}^{\mu ,\nu }\rho ( \hat{a},\hat{d} ) + I_{\hat{b}-,\hat{d}-}^{\mu ,\nu } \rho ( \hat{a},\hat{c} ) \bigr] \\ &\quad \leq \frac{\nu \Gamma ( \mu +1 ) }{2 ( \hat{b}-\hat{a} )^{\mu }} \bigl[ I_{\hat{a}+}^{\mu }\rho ( \hat{b}, \hat{c} ) + I_{\hat{a}+}^{\mu }\rho ( \hat{b}, \hat{d} ) + I_{\hat{b}-}^{\mu }\rho ( \hat{a},\hat{c} ) + I_{\hat{b}-}^{\mu }\rho ( \hat{a},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{1}{\nu +s_{2}}+B(\nu ,s_{2}+1) \biggr\} \\ &\qquad {}+ \frac{\mu \Gamma ( \nu +1 ) }{2 ( \hat{d}-\hat{c} ) ^{\nu }} \bigl[ I_{\hat{c}+}^{\nu } \rho ( \hat{a},\hat{d} ) + I_{\hat{c}+}^{\nu }\rho ( \hat{b},\hat{d} ) + I_{\hat{d}-}^{\nu }\rho ( \hat{a},\hat{c} ) + I_{\hat{d}-}^{\nu }\rho ( \hat{b},\hat{c} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{1}{\mu +s_{1}}+B(\mu ,s_{1}+1) \biggr\} \\ &\quad \leq \mu \nu \bigl[ \rho ( \hat{a},\hat{c} ) +\rho ( \hat{a},\hat{d} ) + \rho ( \hat{b},\hat{c} ) + \rho ( \hat{b},\hat{d} ) \bigr] \\ &\qquad {}\times \biggl\{ \frac{1}{(\mu +s_{1})(\nu +s_{2})}+ \frac{B(\mu ,s_{1}+1)}{\nu +s_{2}}+ \frac{B(\nu ,s_{2}+1)}{\mu +s_{1}} +B( \mu ,s_{1}+1)B(\nu ,s_{2}+1) \biggr\} . \end{aligned}$$
Remark 3.28
If \(\mu =\nu =1\) and \(s_{1}=s_{2}=s\), then the inequalities in Corollary 3.27 coincide with Theorem 2.1 of [1].
