Bivariate Montgomery identity for alpha diamond integrals

Abstract

In the paper, some variants of Montgomery identity with the help of delta and nabla integrals are established which are useful to produce Montgomery identity involving alpha diamond integrals for function of two variables. The aforementioned identity is discussed in discrete, continuous, quantum calculus as well and employed to obtain Ostrowski type inequality for monotonically increasing function with respect to both parameters.

1 Introduction

Mitrinovic, Pecaric, and Fink proved Montgomery identities for functions defined on the real line [17] in the following form:

Let $f: [ {a,b} ] \to \mathbb{R}$ be a differentiable function and $f': [ {a,b} ] \to \mathbb{R}$ be integrable on $[ {a,b} ]\in \mathbb{R}$, then

$$ f ( x ) = \frac{1}{{b - a}} \int _{a}^{b} {f ( t )} \,dt + \int _{a}^{b} {p ( {x,t} )} f' ( t )\,dt, $$

(1.1)

where $p ( {x,t} )$ is called Peano kernel, defined in [17]. Some applications of Montgomery identity in the form of inequalities can be found in [13, 15, 16].

Dragomir et al. extended it for a function of two variables on the real line [12] in the following form:

$$ \begin{aligned}[b] f ( {x,y} ) & = \frac{1}{{b - a}} \int _{a}^{b} {f ( {t,y} )\,dt + } \frac{1}{{d - c}} \int _{c}^{d} {f ( {x,s} )\,ds} \\ &\quad {} - \frac{1}{{ ( {b - a} ) ( {d - c} )}} \int _{a}^{b} { \int _{c}^{d} {f ( {t,s} )\,ds\,dt + } } \int _{a}^{b} { \int _{c}^{d} {p ( {x,t} )q ( {y,s} ) \frac{{{\partial ^{2}}f ( {t,s} )}}{ {\partial t\partial s}}\,ds\,dt} }, \end{aligned} $$

(1.2)

where $f:I = [ {a,b} ] \times [ {c,d} ] \to \mathbb{R}$ is differentiable, the derivative ${\frac{{{\partial ^{2}}f ( {t,s} )}}{{\partial t\partial s}}}$ is integrable on I, $p(x,t)$ and $q(y,s)$ are Peano kernels, defined in [12].

In 1988, Hilger, a German mathematician, presented time scales theory which deals with both discrete and continuous cases simultaneously. Delta and nabla calculus were first two approaches in the theory of time scales. For introduction to the time scales calculus, the readers are referred to [9, 10], and some related inequalities can be seen in [1, 3, 5, 6, 8, 11, 14].

Bohner and Matthews proved the following Montgomery identity for functions of one variable by using delta integrals [7].

Let $a,b,s,t \in \mathbb{T}$; $a < b$ and $f: [ {a,b} ] \to \mathbb{R}$ be differentiable, then

$$ f ( t ) = \frac{1}{{b - a}} \int _{a}^{b} {{f^{ \sigma }} ( s )} \Delta s + \frac{1}{{b - a}} \int _{a}^{b} {p ( {t,s} ){f^{\Delta }} ( s )} \Delta s, $$

(1.3)

where

$$ p ( {t,s} ) = \textstyle\begin{cases} s - a, & a \le s < t, \\ s - b, & t \le s < b. \end{cases} $$

Remark 1.1

For nabla integrals, (1.3) can be written as follows:

Let $a,b,s,t \in \mathbb{T}$; $a < b$ and $f: [ {a,b} ] \to \mathbb{R}$ be differentiable, then

$$ f ( t ) = \frac{1}{{b - a}} \int _{a}^{b} {{f^{ \rho }} ( s )} \nabla s + \frac{1}{{b - a}} \int _{a}^{b} {p ( {t,s} ){f^{\nabla }} ( s )} \nabla s, $$

(1.4)

where

$$ p ( {t,s} ) = \textstyle\begin{cases} s - a, & a < s \le t, \\ s - b, & t < s \le b. \end{cases} $$

Özkan and Yildrim gave the representation of functions depending on two variables in the form of delta integrals [18]. In 2006, Sheng et al. introduced the combined dynamic derivative, also called alpha diamond dynamic derivative $(\alpha \in [0,1])$, as a linear convex combination of the delta and nabla dynamic derivatives on time scales [19]. The aim of the paper is to extend Montgomery identity by using alpha diamond integrals [2] and to establish respective Ostrowski type inequality for alpha diamond integrals.

1.1 Preliminaries

1.1.1 Alpha diamond derivative [19]

Definition 1.2

Let $\mathbb{T}$ be a time scale and $f ( t )$ be differentiable on $t \in \mathbb{T}$ in delta and nabla senses. For $t \in \mathbb{T}$, we define the alpha diamond derivative ${f^{{\diamondsuit _{ \alpha }}}} ( t )$ by

$$ {f^{{\diamondsuit _{\alpha }}}} ( t ) = \alpha {f^{\Delta }} ( t ) + ( {1 - \alpha } ){f^{\nabla }} ( t ) \quad \forall \alpha \in [ {0,1} ] . $$

(1.5)

Note that the alpha diamond derivative reduces to the standard delta derivative for $\alpha = 1$ and nabla derivative for $\alpha = 0$.

Properties of alpha diamond derivative

Theorem 1.3

Let f and g be alpha diamond differentiable functions, then:

The sum $f + g :{\mathbb{T}} \to {\mathbb{R}}$ is alpha diamond differentiable at $t \in {\mathbb{T,}}$ satisfying

$${ ( {f + g} )^{{{\diamondsuit _{\alpha }}}}} ( t ) = {f^{{\diamondsuit _{\alpha }}}} ( t ) + {g^{ {\diamondsuit _{\alpha }}}} ( t ). $$

The product $f g :{\mathbb{T}} \to {\mathbb{R}}$ is alpha diamond differentiable at $t \in {\mathbb{T}}$, satisfying

$${ ( {f g} )^{{{\diamondsuit _{\alpha }}}}} ( t ) = {f^{{\diamondsuit _{\alpha }}}} ( t ).g ( t ) + \alpha {f^{\sigma }} ( t ){g^{\Delta }} ( t ) + ( {1 - \alpha } ){f^{\rho }} ( t ) {g^{\nabla }} ( t ). $$

If $g ( t ){g^{\sigma }} ( t ) {g^{\rho }} ( t ) \ne 0, \frac{f}{g} :{\mathbb{T}} \to {\mathbb{R}}$ is alpha diamond differentiable at $t \in {\mathbb{T,}}$ then

$$ { \biggl( {\frac{f}{g}} \biggr)^{{\diamondsuit _{\alpha }}}} ( t ) = \frac{{{f^{{\diamondsuit _{\alpha }}}} ( t ) {g^{\sigma }} ( t ){g^{\rho }} ( t ) - \alpha {f^{\sigma }} ( t ){g^{\rho }} ( t ){g^{ \Delta }} ( t ) - ( {1 - \alpha } ){f^{ \rho }} ( t ){g^{\sigma }} ( t ){g^{\nabla }} ( t )}}{{g ( t ){g^{\sigma }} ( t ) {g^{\rho }} ( t )}}. $$

1.1.2 Alpha diamond integration [2]

Definition 1.4

Let ${a_{1}}, {a_{2}} \in \mathbb{T}$ and $f: {\mathbb{T}} \to {\mathbb{R}}$, then for $\alpha \in [ {0 , 1}]$, the alpha diamond integral of f is defined by

$$ \int _{{a_{1}}}^{{a_{2}}} {f ( t )} {\diamondsuit _{\alpha }} ( t ) = \alpha \int _{{a_{1}}}^{{a_{2}}} {f ( t )} \Delta t + ( {1 - \alpha } ) \int _{ {a_{1}}}^{{a_{2}}} {f ( t )} \nabla t, $$

provided delta and nabla integrals of f exist on $\mathbb{T}$.

Properties of alpha diamond integration

Theorem 1.5

Let $f, g:{\mathbb{T}} \to {\mathbb{R}}$ be alpha diamond integrable on $[ {a_{1},a_{2}} ]_{\mathbb{T}}$. Let $a_{3} \in { [ {a_{1},a_{2}} ]_{\mathbb{T}}}$ with $a_{1} < a_{3} < a_{2}$ and $\lambda \in {\mathbb{R}}$, then

$$\begin{aligned} & \int _{{a_{1}}}^{{a_{1}}} {f}(t) {\diamondsuit _{\alpha }}t = 0; \\ & \int _{{a_{1}}}^{{a_{2}}} {{f}(t) {\diamondsuit _{\alpha }}t = \int _{{a_{1}}}^{{a_{3}}} {{f}(t) {\diamondsuit _{\alpha }}t + \int _{{a_{3}}}^{{a_{2}}} {{f}(t) {\diamondsuit _{\alpha }} t;} } } \\ & \int _{{a_{1}}}^{{a_{2}}} {f(t)} {\diamondsuit _{\alpha }}t = - \int _{{a_{2}}}^{{a_{1}}} {f(t){\diamondsuit _{\alpha }}t;} \\ & \int _{{a_{1}}}^{{a_{2}}} {({f} + {g}) (t) { \diamondsuit _{ \alpha }}t = \int _{{a_{1}}}^{{a_{2}}} {{f}(t) {\diamondsuit _{ \alpha }}(t) + \int _{{a_{1}}}^{{a_{2}}} {{g}(t) { \diamondsuit _{\alpha }}t;} } } \\ & \int _{{a_{1}}}^{{a_{2}}} \lambda {f}(t) { \diamondsuit _{\alpha }}t = \lambda \int _{{a_{1}}}^{{a_{2}}} {{f}(t) {\diamondsuit _{\alpha }}t.} \end{aligned}$$

The following result can be found in [4] and is used in the proof of next results.

1.1.3 Fubini’s theorem on time scales

Let ${\mathbb{T}_{1}}$, ${\mathbb{T}_{2}}$ be two time scales. Suppose that $S:{\mathbb{T}_{1}} \times {\mathbb{T}_{2}} \to \mathbb{R}$ is integrable with respect to both time scales. Define $\phi ( {{y_{2}}} ) = \int _{{\mathbb{T}_{1}}} {S ( {{y_{1}}, {y_{2}}} )} \Delta {y_{1}}$ for a.e. ${y_{2}} \in {\mathbb{T} _{2}}$ and $\psi ( {{y_{1}}} ) = \int _{{\mathbb{T}_{2}}} {S ( {{y_{1}}, {y_{2}}} )} \Delta {y_{2}}$ for a.e. ${y_{1}} \in {\mathbb{T}_{1}}$. Then ϕ and ψ are both differentiable in time scales settings and

$$ \int _{{\mathbb{T}_{1}}} {\Delta {y_{1}} \int _{{\mathbb{T}_{2}}} {S ( {{y_{1}}, {y_{2}}} )} \Delta {y_{2}} = } \int _{{\mathbb{T}_{2}}} {\Delta {y_{2}} \int _{{\mathbb{T}_{1}}} {S ( {{y_{1}}, {y_{2}}} )} \Delta {y_{1}}}. $$

2 Montgomery identities for function of two variables on time scales

2.1 Montgomery identity I

Lemma 2.1

([18])

Let $m_{1},n_{1} \in {\mathbb{T}_{1}}$, $m_{2},n_{2} \in {\mathbb{T}_{2}}$, and ${f} \in CC_{\mathrm{rd}}^{1} ( {{{ [ {{m_{1}},{n_{1}}} ]}_{{\mathbb{T}_{1}}}} \times {{ [ {{m_{2}},{n_{2}}} ]}_{{\mathbb{T}_{2}}}}, \mathbb{R}} )$, then $\forall ( {x,y} ) \in { [ {{m_{1}},{n_{1}}} ] _{{\mathbb{T}_{1}}}} \times { [ {{m_{2}},{n_{2}}} ]_{ {\mathbb{T}_{2}}}}$, we have the representation

$$ \begin{aligned}[b] f(x,y) & = \frac{1}{k} \biggl[ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl( {{\sigma _{1}} ( s ), {\sigma _{2}} ( t )} \bigr)} } {\Delta _{2}}t{\Delta _{1}}s \\ &\quad {}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )} } \frac{{\partial f ( {{\sigma _{1}} ( s ),t} )}}{{{\Delta _{2}}t}}{\Delta _{2}}t{\Delta _{1}}s \\ & \quad {} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {p ( {x,s} )} } \frac{{\partial f ( {s, {\sigma _{2}} ( t )} )}}{{{\Delta _{1}}s}}{\Delta _{2}}t{\Delta _{1}}s \\ &\quad {}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )q ( {y,t} ) \frac{ {{\partial ^{2}}f ( {s,t} )}}{{{\Delta _{1}}s{\Delta _{2}}t}}} } {\Delta _{2}}t{\Delta _{1}}s \biggr], \end{aligned} $$

(2.1)

where ${p}: [ {{m_{1}},{n_{1}}} ] \times [ {{m_{1}}, {n_{1}}} ] \to \mathbb{R}$ and ${q}: [ {{m_{2}},{n_{2}}} ] \times [ {{m_{2}},{n_{2}}} ] \to \mathbb{R}$ are defined as follows:

$$ {p}(x,s) = \textstyle\begin{cases} s - {m_{1}},& \textit{if }s \in [{m_{1}},x], \\ s - {n_{1}},& \textit{if }s \in (x,{n_{1}}], \end{cases}\displaystyle \qquad {q}(y,t) = \textstyle\begin{cases} t - {m_{2}}, & \textit{if }t \in [{m_{2}},y], \\ t - {n_{2}}, & \textit{if }t \in (y,{n_{2}}], \end{cases} $$

and $k = (n_{1}-m_{1})(n_{2}-m_{2})$.

Note

Throughout the paper, $p(x,s)$, $q(y,t)$, and k are as defined in Lemma 2.1.

2.2 Montgomery identity II

Lemma 2.2

Let $m_{1},n_{1} \in {\mathbb{T}_{1}}$, $m_{2},n_{2} \in {\mathbb{T} _{2}}$, and ${f} \in C{C^{1}} ( {{{ [ {{m_{1}},{n_{1}}} ]} _{{\mathbb{T}_{1}}}} \times {{ [ {{m_{2}},{n_{2}}} ]}_{ {\mathbb{T}_{2}}}},\mathbb{R}} )$, then $\forall ( {x,y} ) \in { [ {{m_{1}},{n_{1}}} ]_{{\mathbb{T}_{1}}}} \times { [ {{m_{2}},{n_{2}}} ]_{{\mathbb{T}_{2}}}}$, we have the representation

$$ \begin{aligned}[b] f ( {x,y} ) &= \frac{1}{k} \biggl[ { \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl( {{\sigma _{1}} ( s ),{\rho _{2}} ( t )} \bigr)} } {\nabla _{2}}t{\Delta _{1}}s} \\ &\quad {}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} ){f^{{\Delta _{1}}}} \bigl( {s,{\rho _{2}} ( t )} \bigr)} } {\nabla _{2}}t { \Delta _{1}}s \\ &\quad {} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} ){f^{{\nabla _{2}}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr)} } {\nabla _{2}}t{ \Delta _{1}}s \\ &\quad {}+ { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )q ( {y,t} ){f^{{\nabla _{2}}{\Delta _{1}}}} ( {s,t} )} } {\nabla _{2}}t{\Delta _{1}}s} \biggr]. \end{aligned} $$

(2.2)

Proof

Apply (1.3) to $f(\cdot ,y)$ for fixed $y \in { [ {{m_{2}}, {n_{2}}} ]_{{\mathbb{T}_{2}}}}$ to get

$$ f ( {x,y} ) = \frac{1}{{n_{1} - m_{1}}} \int _{{m _{1}}}^{{n_{1}}}{f \bigl( {{\sigma _{1}} ( s ),y} \bigr) {\Delta _{1}}s + \frac{1}{{n_{1} - m_{1}}} \int _{{m_{1}}}^{{n _{1}}}{p ( {x,s} ){f^{{\Delta _{1}}}} ( {s,y} ) {\Delta _{1}}s} }. $$

(2.3)

Apply (1.4) to $f(s,\cdot )$ for fixed $s \in { [ {{m_{1}}, {n_{1}}} ]_{{\mathbb{T}_{1}}}}$ to obtain

$$ f ( {s,y} ) = \frac{1}{{n_{2} - m_{2}}} \int _{{m _{2}}}^{{n_{2}}}{f \bigl( {s,{\rho _{2}} ( t )} \bigr)} {\nabla _{2}}t + \frac{1}{{n_{2} - m_{2}}} \int _{{m_{2}}}^{{n _{2}}}{q ( {y,t} ){f^{{\nabla _{2}}}} ( {s,t} ) {\nabla _{2}}t}. $$

(2.4)

Again apply (1.4) to ${f^{{\Delta _{1}}}} ( {s,\cdot } )$ for $s \in { [ {{m_{1}},{n_{1}}} ]_{{\mathbb{T}_{1}}}}$ to get

$$ {f^{{\Delta _{1}}}} ( {s,y} ) = \frac{1}{{n_{2} - m_{2}}} \int _{{m_{2}}}^{{n_{2}}}{{f^{{\Delta _{1}}}} \bigl( {s,{ \rho _{2}} ( t )} \bigr)} {\nabla _{2}}t + \frac{1}{{n_{2} - m _{2}}} \int _{{m_{2}}}^{{n_{2}}}{q ( {y,t} )} {f^{ {\nabla _{2}}{\Delta _{1}}}} ( {s,t} ){\nabla _{2}}t. $$

(2.5)

Substitute (2.4) and (2.5) in (2.3), then use Fubini’s theorem to obtain

$$ \begin{aligned} {f} ( {x,y} ) & = \frac{1}{{{n_{1}} - {m_{1}}}} \int _{{m_{1}}}^{{n_{1}}} \biggl[ { \frac{1}{{{n_{2}} - {m_{2}}}} \int _{{m_{2}}}^{{n_{2}}} {{f} \bigl( {{\sigma _{1}} ( s ), {\rho _{2}} ( t )} \bigr)} {\nabla _{2}}t} \\ &\quad {}+ \frac{1}{{{n_{2}} - {m_{2}}}} \int _{{m_{2}}}^{{n _{2}}} {{q} ( {y,t} )f^{{\nabla _{2}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr)} {\nabla _{2}}t \biggr]{\Delta _{1}}s \\ &\quad {} + \frac{1}{{{n_{1}} - {m_{1}}}} \int _{{m_{1}}}^{{n_{1}}} {p} ( {x,s} ) \biggl[ { \frac{1}{{{n_{2}} - {m_{2}}}} \int _{{m_{2}}}^{{n_{2}}} {f^{{\Delta _{1}}} \bigl( {s,{ \rho _{2}} ( t )} \bigr){\nabla _{2}}t} } \\ &\quad {} + \frac{1}{ {{n_{2}} - {m_{2}}}} \int _{{m_{2}}}^{{n_{2}}} {{q} ( {y,t} )} f^{{\nabla _{2}}{\Delta _{1}}} ( {s,t} ){\nabla _{2}}t \biggr]{\Delta _{1}}s, \end{aligned} $$

which gives the required result. □

2.3 Montgomery identity III

Lemma 2.3

Let $m_{1},n_{1} \in {\mathbb{T}_{1}}$, $m_{2},n_{2} \in {\mathbb{T} _{2}}$, and ${f} \in C{C^{1}} ( {{{ [ {{m_{1}},{n_{1}}} ]} _{{\mathbb{T}_{1}}}} \times {{ [ {{m_{2}},{n_{2}}} ]}_{ {\mathbb{T}_{2}}}}, \mathbb{R}} )$, then $\forall ( {x,y} ) \in { [ {{m_{1}},{n_{1}}} ]_{{\mathbb{T} _{1}}}} \times { [ {{m_{2}},{n_{2}}} ]_{{\mathbb{T}_{2}}}}$, we have the representation

$$ \begin{aligned}[b] f(x,y) & = \frac{{{1}}}{{{k}}} \biggl[ \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{{f}} \bigl( {{\rho _{1}} ( s ),{\sigma _{2}} ( t )} \bigr)} } {\Delta _{2}}t{\nabla _{1}}s \\ &\quad {}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{{p}} ( {{{x,s}}} ) {{{f}}^{{\nabla _{1}}}} \bigl( {s,{\sigma _{2}} ( t )} \bigr)} } { \Delta _{2}}t{\nabla _{1}}s \\ &\quad {} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {{{q}} ( {{{y,t}}} ){{{f}} ^{{\Delta _{2}}}} \bigl( {{\rho _{1}} ( s ),t} \bigr)} } {\Delta _{2}}t{\nabla _{1}}s \\ &\quad {}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{{p}} ( {{{x,s}}} ) {{q}} ( {{{y,t}}} ){{{f}}^{{\nabla _{1}} {\Delta _{2}}}} ( {s,t} )} } { \Delta _{2}}t{ \nabla _{1}}s \biggr]. \end{aligned} $$

(2.6)

Proof

It can be proved accordingly, by shuffling the roles of delta and nabla integrals in Lemma 2.2. □

2.4 Montgomery identity IV

Lemma 2.4

Let $m_{1},n_{1} \in {\mathbb{T}_{1}}$, $m_{2},n_{2} \in {\mathbb{T} _{2}}$, and ${f}, {g} \in CC_{\mathrm{ld}}^{1} ( {{{ [ {{m_{1}}, {n_{1}}} ]}_{{\mathbb{T}_{1}}}} \times {{ [ {{m_{2}}, {n_{2}}} ]}_{{\mathbb{T}_{2}}}},\mathbb{R}} )$, then $\forall ( {x,y} ) \in { [ {{m_{1}},{n_{1}}} ] _{{\mathbb{T}_{1}}}} \times { [ {{m_{2}},{n_{2}}} ]_{ {\mathbb{T}_{2}}}}$, we have the representation

$$\begin{aligned} f(x,y) & = \frac{1}{k} \biggl[ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl( {{\rho _{1}} ( s ), {\rho _{2}} ( t )} \bigr)} } {\nabla _{2}}t{\nabla _{1}}s \\ &\quad {}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )} } \frac{{\partial f ( {{\rho _{1}} ( s ),t} )}}{{{\nabla _{2}}t}}{\nabla _{2}}t{\nabla _{1}}s \\ &\quad {} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {p ( {x,s} )} } \frac{{\partial f ( {s, {\rho _{2}} ( t )} )}}{{{\nabla _{1}}s}}{\nabla _{2}}t {\nabla _{1}}s \\ &\quad {}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {p ( {x,s} )q ( {y,t} ) \frac{{{\partial ^{2}}f ( {s,t} )}}{{{\nabla _{1}}s{\nabla _{2}}t}}} } {\nabla _{2}}t {\nabla _{1}}s \biggr]. \end{aligned}$$

(2.7)

Proof

It can be proved with the help of (1.4) and nabla derivatives and integrals with respect to both variables, instead of (1.3), accordingly as Lemma 2.2. □

2.5 Montgomery identity for alpha diamond integrals

Theorem 2.5

Let $m_{1},n_{1} \in {\mathbb{T}_{1}}$, $m_{2},n_{2} \in {\mathbb{T} _{2}}$, and ${f} \in C{C^{1}} ( {{{ [ {{m_{1}},{n_{1}}} ]} _{{\mathbb{T}_{1}}}} \times {{ [ {{m_{2}},{n_{2}}} ]}_{ {\mathbb{T}_{2}}}}, \mathbb{R}} )$, then $\forall ( {x,y} ) \in { [ {{m_{1}},{n_{1}}} ]_{{\mathbb{T} _{1}}}} \times { [ {{m_{2}},{n_{2}}} ]_{{\mathbb{T}_{2}}}}$, we have

$$ \begin{aligned}[b] k{f}(x,y) &= \biggl[ {\alpha _{1}} {\alpha _{2}} \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{f} \bigl({\sigma _{1}}(s), {\sigma _{2}}(t) \bigr){\Delta _{2}}t {\Delta _{1}}s} } \\ &\quad {}+ {\alpha _{1}}(1 - {\alpha _{2}}) \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{f} \bigl({\sigma _{1}}(s), {\rho _{2}}(t) \bigr){\nabla _{2}}t {\Delta _{1}}s} } \\ &\quad {} + {\alpha _{2}}(1 - {\alpha _{1}}) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{f} \bigl({\rho _{1}}(s),{\sigma _{2}}(t) \bigr) {\Delta _{2}}t {\nabla _{1}}s} } \\ &\quad {}+ (1 - {\alpha _{1}}) (1 - {\alpha _{2}}) { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{f} \bigl({\rho _{1}}(s),{\rho _{2}}(t) \bigr){\nabla _{2}}t {\nabla _{1}}s} } } \biggr] \\ &\quad {} + \biggl[ {{\alpha _{2}} \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p}(x,s)f^{{\diamondsuit _{{\alpha _{1}}}}} \bigl(s, {\sigma _{2}}(t) \bigr){\Delta _{2}}t { \diamondsuit _{{\alpha _{1}}}}s} } } \\ &\quad {}+ (1 - {\alpha _{2}}) \int _{{m_{1}}}^{{n _{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p}(x,s)f^{{\diamondsuit _{{\alpha _{1}}}}} \bigl(s, {\rho _{2}}(t) \bigr){\nabla _{2}}t { \diamondsuit _{{\alpha _{1}}}}s} } \biggr] \\ &\quad {} + \biggl[ {\alpha _{1}} \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{q}(y,t)f^{{\diamondsuit _{\alpha 2}}} \bigl( {\sigma _{1}}(s),t \bigr){\Delta _{2}}t{ \diamondsuit _{{\alpha _{1}}}}s} } \\ &\quad {}+ (1 - {\alpha _{1}}) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{q}(y,t)f^{{\diamondsuit _{{\alpha _{2}}}}} \bigl({\rho _{1}}(s),t \bigr){\nabla _{2}}t { \diamondsuit _{{\alpha _{1}}}}s} } \biggr] \\ &\quad {} + \int _{{m_{1}}}^{{n_{1}}} \int _{{m_{2}}}^{{n_{2}}} {{p}(x,s){q}(y,t)f^{{\diamondsuit _{{\alpha _{1}}}}{\diamondsuit _{\alpha _{2}}}}(s,t) {\diamondsuit _{{\alpha _{2}}}}t {\diamondsuit _{{\alpha _{1}}}}s}. \end{aligned} $$

(2.8)

Proof

Multiply (2.1) by ${\alpha _{1}}{\alpha _{2}}$, (2.2) by ${\alpha _{1}} ( {1 - {\alpha _{2}}} )$, (2.6) by ${\alpha _{2}} ( {1 - {\alpha _{1}}} )$, and (2.7) by $( {1 - {\alpha _{1}}} ) ( {1 - {\alpha _{2}}} )$, then add the resultants to obtain

$$\begin{aligned} k{f}(x,y) =& {\alpha _{1}} {\alpha _{2}} \biggl[ \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{f} \bigl( {{\sigma _{1}} ( s ),{\sigma _{2}} ( t )} \bigr)} } {\Delta _{2}}t{\Delta _{1}}s \\ &{}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p} ( {x,s} )f^{{\Delta _{1}}} \bigl( {s,{\sigma _{2}} ( t )} \bigr)} } {\Delta _{2}}t { \Delta _{1}}s \\ &{} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )f^{{\Delta _{2}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr)} } {\Delta _{2}}t{ \Delta _{1}}s \\ &{}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )q ( {y,t} )f^{{\Delta _{1}}{\Delta _{2}}} ( {s,t} )} } {\Delta _{2}}t{\Delta _{1}}s \biggr] \\ &{} + {\alpha _{1}} ( {1 - {\alpha _{2}}} ) \biggl[ { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{f} \bigl( {{\sigma _{1}} ( s ),{\rho _{2}} ( t )} \bigr)} } {\nabla _{2}}t{\Delta _{1}}s} \\ &{}+ \int _{{m _{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p} ( {x,s} )f ^{{\Delta _{1}}} \bigl( {s,{\rho _{2}} ( t )} \bigr)} } {\nabla _{2}}t{ \Delta _{1}}s \\ &{} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )f^{{\nabla _{2}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr)} } {\nabla _{2}}t{ \Delta _{1}}s \\ &{}+ { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )q ( {y,t} )f^{{\Delta _{1}}{\nabla _{2}}} ( {s,t} )} } {\nabla _{2}}t{\Delta _{1}}s} \biggr] \\ &{} + {\alpha _{2}} ( {1 - {\alpha _{1}}} ) \biggl[ { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{f} \bigl( {{\rho _{1}} ( s ),{\sigma _{2}} ( t )} \bigr)} } {\Delta _{2}}t{\nabla _{1}}} s\biggr] \\ &{}+ \int _{{m _{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )f ^{{\nabla _{1}}} \bigl( {s,{\sigma _{2}} ( t )} \bigr)} } {\Delta _{2}}t{ \nabla _{1}}s \\ &{} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )f^{{\Delta _{2}}} \bigl( {{\rho _{1}} ( s ),t} \bigr)} } {\Delta _{2}}t{ \nabla _{1}}s \\ &{}+ \int _{ {m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )q ( {y,t} )f^{{\nabla _{1}}{\Delta _{2}}} ( {s,t} )} } {\Delta _{2}}t{\nabla _{1}}s \\ &{} + ( {1 - {\alpha _{1}}} ) ( {1 - {\alpha _{2}}} ) \biggl[ { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{ {n_{2}}} {{f} \bigl( {{\rho _{1}} ( s ),{\rho _{2}} ( t )} \bigr)} } {\nabla _{2}}t{\nabla _{1}}s} \\ &{}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p} ( {x,s} )f^{{\nabla _{1}}} \bigl( {s,{\rho _{2}} ( t )} \bigr)} } {\nabla _{2}}t{ \nabla _{1}}s \\ &{} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )f^{{\nabla _{2}}} \bigl( {{\rho _{1}} ( s ),t} \bigr)} } {\nabla _{2}}t{ \nabla _{1}}s \\ &{}+ { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )q ( {y,t} )f^{{\nabla _{1}}{\nabla _{2}}} ( {s,t} )} } {\nabla _{2}}t{\nabla _{1}}s} \biggr] \\ = & \biggl[ {{\alpha _{1}} {\alpha _{2}} \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl({\sigma _{1}}(s),{\sigma _{2}}(t) \bigr) {\Delta _{2}}t {\Delta _{1}}s} } } \\ &{}+{\alpha _{1}}(1 - {\alpha _{2}}) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {f \bigl({\sigma _{1}}(s),{\rho _{2}}(t) \bigr){\nabla _{2}}t {\Delta _{1}}s} } \\ &{}+{\alpha _{2}}(1 - {\alpha _{1}}) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl({\rho _{1}}(s),{\sigma _{2}}(t) \bigr) {\Delta _{2}}t {\nabla _{1}}s} } \\ &{}+ (1 - { \alpha _{1}}) (1 - {\alpha _{2}}) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {f \bigl({\rho _{1}}(s),{\rho _{2}}(t) \bigr){\nabla _{2}}t {\nabla _{1}}s} } \biggr] \\ &{} + \biggl[ {{\alpha _{1}} {\alpha _{2}} \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p}(x,s)f^{{\Delta _{1}}} \bigl(s,{\sigma _{2}}(t) \bigr){\Delta _{2}}t { \Delta _{1}}s} } } \\ &{}+ {\alpha _{1}}(1 - {\alpha _{2}}) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {{p}(x,s)f^{{\Delta _{1}}} \bigl(s,{\rho _{2}}(t) \bigr){\nabla _{2}}t { \Delta _{1}}s} } \\ &{} + {\alpha _{2}}(1 - {\alpha _{1}}) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p}(x,s)f^{{\nabla _{1}}} \bigl(s,{\sigma _{2}}(t) \bigr){\Delta _{2}}t { \nabla _{1}}s} } \\ &{}+ ( {1 - {\alpha _{1}}} ) ( {1 - {\alpha _{2}}} ) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p} ( {x,s} )f^{{\nabla _{1}}} \bigl( {s,{\rho _{2}} ( t )} \bigr)} } {\nabla _{2}}t{ \nabla _{1}}s \biggr] \\ &{} + \biggl[ {{\alpha _{1}} {\alpha _{2}} \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{q} ( {y,t} )f^{{\Delta _{2}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr)} } {\Delta _{2}}t{ \Delta _{1}}s} \\ &{}+ {\alpha _{1}} ( {1 - {\alpha _{2}}} ) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n _{2}}} {{q} ( {y,t} )f^{{\nabla _{2}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr)} } {\nabla _{2}}t{ \Delta _{1}}s \\ &{} + {\alpha _{2}} ( {1 - {\alpha _{1}}} ) \int _{ {m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{q} ( {y,t} )f^{{\Delta _{2}}} \bigl( {{\rho _{1}} ( s ),t} \bigr)} } {\Delta _{2}}t{ \nabla _{1}}s \\ &{}+ ( {1 - {\alpha _{1}}} ) ( {1 - {\alpha _{2}}} ) \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{q} ( {y,t} )f ^{{\nabla _{2}}} \bigl( {{\rho _{1}} ( s ),t} \bigr)} } {\nabla _{2}}t{ \nabla _{1}}s \biggr] \\ &{} + \biggl[ {{\alpha _{1}} {\alpha _{2}} \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p} ( {x,s} ){q} ( {y,t} )f^{{\Delta _{1}}{\Delta _{2}}} ( {s,t} )} } {\Delta _{2}}t} {\Delta _{1}}s \\ &{}+ {\alpha _{1}} ( {1 - {\alpha _{2}}} ) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {{p} ( {x,s} ){q} ( {y,t} )f^{ {\Delta _{1}}{\nabla _{2}}} ( {s,t} )} } {\nabla _{2}}t{\Delta _{1}}s \\ &{} + {\alpha _{2}} ( {1 - {\alpha _{1}}} ) \int _{ {m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p} ( {x,s} ){q} ( {y,t} )f^{{\nabla _{1}}{\Delta _{2}}} ( {s,t} )} } {\Delta _{2}}t{\nabla _{1}}s \\ &{} + ( {1 - {\alpha _{1}}} ) ( {1 - {\alpha _{2}}} ) { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{ {n_{2}}} {{p} ( {x,s} ){q} ( {y,t} )f^{{\nabla _{1}}{\nabla _{2}}} ( {s,t} )} } {\nabla _{2}}t{\nabla _{1}}s} \biggr], \end{aligned}$$

and after simplification one gets the required result. □

Example 2.6

For $\mathbb{T}_{1}=h_{1}\mathbb{N}$, $\mathbb{T}_{2}=h_{2}\mathbb{N}$, $h _{1},h_{2}>0$, (2.8) can be written as follows:

$$\begin{aligned} k{f} ( {x,y} ) =& {\alpha _{1}} {\alpha _{2}}\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}}}^{ \frac{{{n_{1}}}}{{{h_{1}}}} - 1} {\sum _{j = \frac{{{m_{2}}}}{{{h_{2}}}}}^{ \frac{{{n_{2}}}}{{{h_{2}}}} - 1} {{f} \bigl( { ( {i + 1} ) {h_{1}}, ( {i + 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } \\ &{} + {\alpha _{1}}(1 - {\alpha _{2}})\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}}}^{ \frac{{{n_{1}}}}{{{h_{1}}}} - 1} {\sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}} + 1}^{\frac{{{n_{2}}}}{{{h _{2}}}}} {{f} \bigl( { ( {i + 1} ){h_{1}}, ( {i - 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } \\ &{} + {\alpha _{2}}(1 - {\alpha _{1}})\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}} + 1}^{\frac{{{n_{1}}}}{{{h _{1}}}}} { \sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}}}^{\frac{{{n _{2}}}}{{{h_{2}}}} - 1} {{f} \bigl( { ( {i - 1} ){h_{1}}, ( {i + 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } \\ &{} + (1 - {\alpha _{1}}) (1 - {\alpha _{2}})\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}} + 1}^{\frac{{{n_{1}}}}{{{h _{1}}}}} {\sum _{j = \frac{{{m_{2}}}}{{{h_{2}}}} + 1}^{\frac{ {{n_{2}}}}{{{h_{2}}}}} {{f} \bigl( { ( {i - 1} ){h_{1}}, ( {i - 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } \\ &{} + {\alpha _{2}} \Biggl[ {{\alpha _{1}}\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}}}^{ \frac{{{n_{1}}}}{{{h_{1}}}} - 1} {\sum _{j = \frac{{{m_{2}}}}{{{h_{2}}}}}^{ \frac{{{n_{2}}}}{{{h_{2}}}} - 1} {p(x,{i + 1})f ^{\diamondsuit {\alpha _{1}}} \bigl( { ( {i + 1} ) {h_{1}}, ( {i + 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } } \\ &{} + (1 - {\alpha _{1}})\sum_{i = \frac{{{m_{1}}}}{{{h_{1}}}} + 1}^{\frac{{{n_{1}}}}{{{h _{1}}}}} {\sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}}}^{\frac{{{n _{2}}}}{{{h_{2}}}} - 1} {p(x,{i + 1}) f ^{\diamondsuit {\alpha _{1}}} \bigl( { ( {i - 1} ) {h_{1}}, ( {i + 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } \Biggr] \\ &{} + (1 - {\alpha _{2}}) \Biggl[ {{\alpha _{1}} \sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}}}^{ \frac{{{n_{1}}}}{{{h_{1}}}} - 1} {\sum _{j = \frac{{{m_{2}}}}{{{h_{2}}}} + 1}^{\frac{{{n_{2}}}}{{{h _{2}}}}} {p(x,{i - 1})f ^{\diamondsuit {\alpha _{1}}} \bigl( { ( {i + 1} ){h_{1}}, ( {i - 1} ) {h_{2}}} \bigr){h_{1}} {h_{2}}} } } \\ &{} + (1 - {\alpha _{1}})\sum_{i = \frac{{{m_{1}}}}{{{h_{1}}}} + 1}^{\frac{{{n_{1}}}}{{{h _{1}}}}} {\sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}} + 1}^{\frac{ {{n_{2}}}}{{{h_{2}}}}} {p(x,{i - 1}) f ^{\diamondsuit {\alpha _{1}}} \bigl( { ( {i - 1} ) {h_{1}}, ( {i - 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } \Biggr] \\ &{} + {\alpha _{1}} \Biggl[ {{\alpha _{2}}\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}}}^{ \frac{{{n_{1}}}}{{{h_{1}}}} - 1} {\sum _{j = \frac{{{m_{2}}}}{{{h_{2}}}}}^{ \frac{{{n_{2}}}}{{{h_{2}}}} - 1} {q(y,{i + 1})f ^{\diamondsuit {\alpha _{2}}} \bigl( { ( {i + 1} ) {h_{1}}, ( {i + 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } } \\ &{} + (1 - {\alpha _{2}})\sum_{i = \frac{{{m_{1}}}}{{{h_{1}}}}}^{ \frac{{{n_{1}}}}{{{h_{1}}}} - 1} {\sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}} + 1}^{\frac{{{n_{2}}}}{{{h _{2}}}}} {q(y,{i + 1}) f ^{\diamondsuit {\alpha _{2}}} \bigl( { ( {i + 1} ){h_{1}}, ( {i - 1} ) {h_{2}}} \bigr){h_{1}} {h_{2}}} } \Biggr] \\ &{} + (1 - {\alpha _{1}}) \Biggl[ {{\alpha _{2}} \sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}} + 1}^{\frac{{{n_{1}}}}{{{h _{1}}}}} {\sum _{j = \frac{{{m_{2}}}}{{{h_{2}}}}}^{\frac{{{n _{2}}}}{{{h_{2}}}} - 1} {q(y,{i - 1})f ^{\diamondsuit {\alpha _{2}}} \bigl( { ( {i - 1} ) {h_{1}}, ( {i + 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } } \\ &{} + (1 - {\alpha _{2}})\sum_{i = \frac{{{m_{1}}}}{{{h_{1}}}} + 1}^{\frac{{{n_{1}}}}{{{h _{1}}}}} {\sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}} + 1}^{\frac{ {{n_{2}}}}{{{h_{2}}}}} {q(y,{i - 1}) f ^{\diamondsuit {\alpha _{2}}} \bigl( { ( {i - 1} ) {h_{1}}, ( {i - 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } \Biggr] \\ &{} + {\alpha _{1}} {\alpha _{2}}\sum _{i = \frac{{{m_{1}}}}{{{h _{1}}}}}^{\frac{{{n_{1}}}}{{{h_{1}}}} - 1} {\sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}}}^{ \frac{{{n_{2}}}}{{{h_{2}}}} - 1} {p(x,{i + 1})q(y,{i + 1})f ^{\diamondsuit {\alpha _{1}}\diamondsuit {\alpha _{2}}} \bigl( { ( {i + 1} ){h_{1}}, ( {i + 1} ){h_{2}}} \bigr) {h_{1}} {h_{2}}} } \\ &{} + {\alpha _{1}}(1 - {\alpha _{2}})\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}}}^{ \frac{{{n_{1}}}}{{{h_{1}}}} - 1} {\sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}} + 1}^{\frac{{{n_{2}}}}{{{h _{2}}}}} {p(x,{i + 1})q(y,{i - 1})f ^{\diamondsuit {\alpha _{1}}\diamondsuit {\alpha _{2}}} \bigl( { ( {i + 1} ){h_{1}}, ( {i - 1} ){h_{2}}} \bigr) {h_{1}} {h_{2}}} } \\ &{} + {\alpha _{2}}(1 - {\alpha _{1}})\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}} + 1}^{\frac{{{n_{1}}}}{{{h _{1}}}}} {\sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}}}^{\frac{{{n _{2}}}}{{{h_{2}}}} - 1} {p(x,{i - 1})q(y,{i + 1})f ^{\diamondsuit {\alpha _{1}}\diamondsuit {\alpha _{2}}} \bigl( { ( {i - 1} ){h_{1}}, ( {i + 1} ){h_{2}}} \bigr) {h_{1}} {h_{2}}} } \\ &{} + (1 - {\alpha _{1}}) (1 - {\alpha _{2}})\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}} + 1}^{\frac{{{n_{1}}}}{{{h _{1}}}}} \sum _{j = \frac{{{m_{2}}}}{{{h_{2}}}} + 1}^{\frac{ {{n_{2}}}}{{{h_{2}}}}} p(x,{i - 1})q(y,{i - 1}) \\ &{}\times f ^{\diamondsuit {\alpha _{1}}\diamondsuit {\alpha _{2}}} \bigl( { ( {i - 1} ){h_{1}}, ( {i - 1} ){h_{2}}} \bigr) {h_{1}} {h_{2}} . \end{aligned}$$

Example 2.7

If ${\mathbb{T}_{1}} = q_{1}^{\mathbb{N}}$, ${\mathbb{T}_{2}} = q_{2} ^{\mathbb{N}}$, ${q_{1}},{q_{2}} > 1$, and $a_{1} = q_{1}^{{m_{1}}}$, $b_{1} = q_{1}^{{n_{1}}}$, $c_{1} = q_{2}^{{m_{2}}}$, and $d_{1} = q _{2}^{{n_{2}}}$, then for ${m_{1}}, {m_{2}}, {n_{1}}, {n_{2}} \in \mathbb{ N}$, (2.8) takes the form

$$\begin{aligned}& \bigl( {q_{1}^{{n_{1}}} - q_{1}^{{m_{1}}}} \bigr) \bigl( {q_{2} ^{{n_{2}}} - q_{2}^{{m_{2}}}} \bigr){f}(x,y) \\& \quad = ({q_{1}} - 1) ( {q_{2}} - 1) \Biggl[ {{\alpha _{1}} {\alpha _{2}}\sum_{{k_{1}} = {m_{1}}}^{{n_{1}} - 1} {\sum_{{k_{2}} = {m_{2}}}^{{n_{2}} - 1} {{f} \bigl(q_{1}^{{k_{1}} + 1},q_{2}^{{k_{2}} + 1}} } \bigr)} q_{1}^{{k_{1}}}q_{2}^{{k_{2}}} \\& \qquad {} + {\alpha _{1}} ( {1 - {\alpha _{2}}} )\sum _{{k_{1}} = {m_{1}}}^{{n_{1}} - 1} {\sum _{{k_{2}} = {m_{2}} + 1}^{{n_{2}}} {{f} \bigl(q_{1}^{{k_{1}} + 1},} } q_{2}^{{k_{2}} - 1} \bigr)q_{1}^{{k_{1}}}q_{2}^{{k_{2}} - 1} \\& \qquad {} + {\alpha _{2}} ( {1 - {\alpha _{1}}} )\sum _{{k_{1}} = {m_{1}} + 1}^{{n_{1}}} {\sum _{{k_{2}} = {m_{2}}}^{{n_{2}} - 1} {{f} \bigl(q_{1}^{{k_{1}} - 1},} } q_{2}^{{k_{2}} + 1} \bigr)q_{1}^{{k_{1}} - 1}q_{2}^{{k_{2}}} \\& \qquad {} + ( {1 - {\alpha _{1}}} ) ( {1 - {\alpha _{2}}} ) \sum_{{k_{1}} = {m_{1}} + 1}^{{n_{1}}} { \sum_{{k_{2}} = {m_{2}} + 1}^{{n_{2}}} {{f} \bigl(q_{1}^{{k_{1}} - 1},} } q_{2}^{{k_{2}} - 1} \bigr)q_{1}^{{k_{1}} - 1}q_{2}^{{k_{2}} - 1} \\& \qquad {} + {\alpha _{2}} \Biggl\{ {{\alpha _{1}}\sum _{{k_{1}} = {m_{1}}} ^{{n_{1}} - 1} {\sum _{{k_{2}} = {m_{2}}}^{{n_{2}} - 1} {p \bigl(x,q _{2}^{{k_{2}} + 1} \bigr){{f}^{\diamondsuit {\alpha _{1}}}} \bigl(q_{1}^{{k_{1}} + 1},} } q_{2}^{{k_{2}} + 1} \bigr)} q_{1}^{{k_{1}}}q_{2}^{{k_{2}}} \\& \qquad {}+ {(1 - {\alpha _{1}})\sum_{{k_{1}} = {m_{1}} + 1} ^{{n_{1}}} {\sum_{{k_{2}} = {m_{2}}}^{{n_{2}} - 1} {p \bigl(x,q_{2} ^{{k_{2}} + 1} \bigr){{f}^{\diamondsuit {\alpha _{1}}}} \bigl(q_{1}^{{k_{1}} - 1},} } q_{2}^{{k_{2}} + 1} \bigr)q_{1}^{{k_{1}} - 1}q_{2}^{{k_{2}}}} \Biggr\} \\& \qquad {} + (1 - {\alpha _{2}}) \Biggl\{ {{\alpha _{1}} \sum_{{k_{1}} = {m_{1}}}^{{n_{1}} - 1} {\sum _{{k_{2}} = {m_{2}} + 1}^{{n_{2}}} {p \bigl(x,q_{2}^{{k_{2}} - 1} \bigr) {{f}^{\diamondsuit {\alpha _{1}}}} \bigl(q_{1}^{{k_{1}} + 1},} } q_{2}^{ {k_{2}} - 1} \bigr)} q_{1}^{{k_{1}}}q_{2}^{{k_{2}} - 1} \\& \qquad {} + {(1 - {\alpha _{1}})\sum_{{k_{1}} = {m_{1}} + 1} ^{{n_{1}}} {\sum_{{k_{2}} = {m_{2}} + 1}^{{n_{2}}} {p \bigl(x,q_{2} ^{{k_{2}} - 1} \bigr){{f}^{\diamondsuit {\alpha _{1}}}} \bigl(q_{1}^{{k_{1}} - 1},} } q_{2}^{{k_{2}} - 1} \bigr)q_{1}^{{k_{1}} - 1}q_{2}^{{k_{2}} - 1}} \Biggr\} \\& \qquad {} + {\alpha _{1}} \Biggl\{ {{\alpha _{2}}\sum _{{k_{1}} = {m_{1}}} ^{{n_{1}} - 1} {\sum _{{k_{2}} = {m_{2}}}^{{n_{2}} - 1} {q \bigl(y,q _{2}^{{k_{2}} + 1} \bigr){{f}^{\diamondsuit {\alpha _{2}}}} \bigl(q_{1}^{{k_{1}} + 1},} } q_{2}^{{k_{2}} + 1} \bigr)} q_{1}^{{k_{1}}}q_{2}^{{k_{2}}} \\& \qquad {} + {(1 - {\alpha _{2}})\sum_{{k_{1}} = {m_{1}}}^{ {n_{1}} - 1} {\sum_{{k_{2}} = {m_{2}} + 1}^{{n_{2}}} {q \bigl(y,q _{2}^{{k_{2}} + 1} \bigr){{f}^{\diamondsuit {\alpha _{2}}}} \bigl(q_{1}^{{k_{1}} + 1},} } q_{2}^{{k_{2}} - 1} \bigr)q_{1}^{{k_{1}}}q_{2}^{{k_{2}} - 1}} \Biggr\} \\& \qquad {} + (1 - {\alpha _{1}}) \Biggl\{ {{\alpha _{2}} \sum_{{k_{1}} = {m_{1}} + 1}^{{n_{1}}} {\sum _{{k_{2}} = {m_{2}}}^{{n_{2}} - 1} {q \bigl(y,q_{2}^{{k_{2}} - 1} \bigr){{f}^{\diamondsuit {\alpha _{2}}}} \bigl(q_{1}^{{k_{1}} - 1},} } q_{2}^{{k_{2}} + 1} \bigr)} q _{1}^{{k_{1}} - 1}q_{2}^{{k_{2}}} \\& \qquad {} + {(1 - {\alpha _{2}})\sum_{{k_{1}} = {m_{1}} + 1} ^{{n_{1}}} {\sum_{{k_{2}} = {m_{2}} + 1}^{{n_{2}}} {q \bigl(y,q_{2} ^{{k_{2}} - 1} \bigr){{f}^{\diamondsuit {\alpha _{2}}}} \bigl(q_{1}^{{k_{1}} - 1},} } q_{2}^{{k_{2}} - 1} \bigr)q_{1}^{{k_{1}} - 1}q_{2}^{{k_{2}} - 1}} \Biggr\} \\& \qquad {} + {\alpha _{1}} {\alpha _{2}}\sum _{{k_{1}} = {m_{1}}}^{{n_{1}} - 1} {\sum_{{k_{2}} = {m_{2}}}^{{n_{2}} - 1} {p \bigl(x,q_{1}^{{k _{1}} + 1} \bigr)q \bigl(y,q_{2}^{{k_{2}} + 1} \bigr){{f}^{\diamondsuit {\alpha _{1}} \diamondsuit {\alpha _{2}}}} \bigl(q_{1}^{{k_{1}} + 1},} } q_{2} ^{{k_{2}} + 1} \bigr)q_{1}^{{k_{1}}}q_{2}^{{k_{2}}} \\& \qquad {} + {\alpha _{1}}(1 - {\alpha _{2}})\sum _{{k_{1}} = {m_{1}}} ^{{n_{1}} - 1} {\sum _{{k_{2}} = {m_{2}} + 1}^{{n_{2}}} {p \bigl(x,q _{1}^{{k_{1}} + 1} \bigr)q \bigl(y,q_{2}^{{k_{2}} - 1} \bigr)){{f}^{\diamondsuit {\alpha _{1}}\diamondsuit {\alpha _{2}}}} \bigl(q_{1}^{{k_{1}} + 1},} } q _{2}^{{k_{2}} - 1} \bigr)q_{1}^{{k_{1}}}q_{2}^{{k_{2}} - 1} \\& \qquad {} + {\alpha _{2}}(1 - {\alpha _{1}})\sum _{{k_{1}} = {m_{1}} + 1} ^{{n_{1}}} { \sum _{{k_{2}} = {m_{2}}}^{{n_{2}} - 1} {p \bigl(x,q_{1} ^{{k_{1}} - 1} \bigr)q \bigl(y,q_{2}^{{k_{2}} + 1} \bigr){{f}^{\diamondsuit {\alpha _{1}} \diamondsuit {\alpha _{2}}}} \bigl(q_{1}^{{k_{1}} - 1},} } q_{2} ^{{k_{2}} + 1} \bigr)q_{1}^{{k_{1}} - 1}q_{2}^{{k_{2}}} \\& \qquad {} + {(1 - {\alpha _{1}}) (1 - {\alpha _{2}}) \sum_{{k_{1}} = {m_{1}} + 1}^{{n_{1}}} {\sum _{{k_{2}} = {m_{2}} + 1}^{{n_{2}}} {p \bigl(x,q_{1}^{{k_{1}} - 1} \bigr)q \bigl(y,q _{2}^{{k_{2}} - 1} \bigr){{f}^{\diamondsuit {\alpha _{1}}\diamondsuit {\alpha _{2}}}} \bigl(q_{1}^{{k_{1}} - 1},} } q_{2}^{{k_{2}} - 1} \bigr)} \Biggr]. \end{aligned}$$

Remark 2.8

For $\mathbb{T}_{1}=\mathbb{T}_{2}=\mathbb{R}$, (2.8) coincides with (1.2).

Corollary 2.9

In addition to the conditions of Theorem 2.5, if the function is monotonically increasing, then we have the following inequality:

$$ \begin{aligned}[b] f ( {x,y} ) &\le \frac{1}{k} \biggl[ { \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl( {{\sigma _{1}} ( s ),{\sigma _{2}} ( t )} \bigr)} } } { \diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s \\ &\quad {}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )} } {} {f^{{\diamondsuit _{{\alpha _{1}}}}}} \bigl( {s,{\sigma _{2}} ( t )} \bigr){ \diamondsuit _{{\alpha _{2}}}}t {\diamondsuit _{{\alpha _{1}}}}s \\ &\quad {} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )} } {f^{{\diamondsuit _{{\alpha _{2}}}}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr){ \diamondsuit _{{\alpha _{2}}}}t {\diamondsuit _{{\alpha _{1}}}}s \\ &\quad {}+ \int _{{m_{1}}}^{ {n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )} } q ( {y,t} ){f^{{\diamondsuit _{{\alpha _{2}}}}{\diamondsuit _{{\alpha _{1}}}}}} ( {s,t} ){\diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s \biggr]. \end{aligned} $$

(2.9)

Proof

Since f is increasing, ${\rho _{i}} ( {{t_{i}}} ) \le {t_{i}} \le {\sigma _{i}} ( {{t_{i}}} )$, $\forall {t_{i}} \in {\mathbb{T}_{i}}$, therefore by replacing ${\rho _{i}}$ with ${\sigma _{i}}$, on the right-hand side of (2.8), one gets the required result. □

3 Ostrowski type inequality

Theorem 3.1

Let $m_{1},n_{1} \in {\mathbb{T}_{1}}$, $m_{2},n_{2} \in {\mathbb{T} _{2}}$, ${f} \in C{C^{1}} ( {{{ [ {{m_{1}},{n_{1}}} ]} _{{\mathbb{T}_{1}}}} \times {{ [ {{m_{2}},{n_{2}}} ]}_{ {\mathbb{T}_{2}}}}, \mathbb{R}} )$, further assume $f(\cdot , \cdot )$ is an increasing function with respect to both parameters, then $\forall ( {x,y} ) \in { [ {{m_{1}},{n_{1}}} ] _{{\mathbb{T}_{1}}}} \times { [ {{m_{2}},{n_{2}}} ]_{ {\mathbb{T}_{2}}}}$, one gets

$$\begin{aligned} & \biggl\vert {f ( {x,y} ) - \frac{1}{k} \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl( {{\sigma _{1}} ( s ),{\sigma _{2}} ( t )} \bigr){\diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s} } } \biggr\vert \\ &\quad \le \frac{1}{{ ( {{n_{1}} - {m_{1}}} )}}{M_{1}} \bigl[ { \hat{{h_{2}}} ( {x,{m_{1}}} ) - \hat{{h_{2}}} ( {x,{n_{1}}} )} \bigr] + \frac{1}{{ ( {{n_{2}} - {m_{2}}} )}}{M_{2}} \bigl[ {\hat{{h_{2}}} ( {y,{m_{2}}} ) - \hat{{h_{2}}} ( {y,{n_{2}}} )} \bigr] \\ &\qquad {}+ \frac{1}{k}{M_{3}} \bigl[ { \hat{{h_{2}}} ( {x,{m_{1}}} ) - \hat{{h_{2}}} ( {x, {n_{1}}} )} \bigr] \bigl[ { \hat{{h_{2}}} ( {y,{m_{2}}} ) - \hat{{h_{2}}} ( {y,{n_{2}}} )} \bigr], \end{aligned}$$

where

$$\begin{aligned}& \hat{{h_{2}}} ( {x,{m_{1}}} ) = \int _{{m_{1}}}^{x} { ( {s - {m_{1}}} )} {\diamondsuit _{{\alpha _{1}}}}s,\qquad \hat{{h_{2}}} ( {x,{n_{1}}} ) = \int _{{n_{1}}}^{x} { ( {s - {n_{1}}} )} {\diamondsuit _{{\alpha _{1}}}}s, \\& \hat{{h_{2}}} ( {y,{m_{2}}} ) = \int _{{m_{2}}}^{y} { ( {t - {m_{2}}} )} {\diamondsuit _{{\alpha _{2}}}}t,\qquad \hat{{h_{2}}} ( {y,{n_{2}}} ) = \int _{{n_{2}}}^{y} { ( {t - {n_{2}}} )} {\diamondsuit _{{\alpha _{2}}}}t. \end{aligned}$$

Proof

Inequality (2.9) can be written as

$$\begin{aligned} &f ( {x,y} ) - \frac{1}{k} \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl( {{\sigma _{1}} ( s ), {\sigma _{2}} ( t )} \bigr)} } { \diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s \\ & \quad \le \frac{1}{k} \biggl[ { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )} } {f^{{\diamondsuit _{{\alpha _{1}}}}}} \bigl( {s,{\sigma _{2}} ( t )} \bigr){ \diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s} \\ &\qquad {} + \int _{{m _{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )} } {f^{{\diamondsuit _{{\alpha _{2}}}}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr){ \diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s \\ & \qquad {} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {p ( {x,s} )} } q ( {y,t} ){f^{ {\diamondsuit _{{\alpha _{2}}}}{\diamondsuit _{{\alpha _{1}}}}s}} ( {s,t} ){\diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s \biggr]. \end{aligned}$$

By taking absolute value on both sides, one gets

$$\begin{aligned} & \biggl\vert {f ( {x,y} ) - \frac{1}{k} \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl( {{\sigma _{1}} ( s ),{\sigma _{2}} ( t )} \bigr){\diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s} } } \biggr\vert \\ &\quad \le \frac{1}{k} \biggl[ {{M_{1}} \biggl\vert { \int _{{m_{1}}}^{ {n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} ) {\diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s} } } \biggr\vert } + {M_{2}} \biggl\vert { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} ){\diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s} } } \biggr\vert \\ &\qquad {} + {M_{3}} \biggl\vert { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )q ( {y,t} ) { \diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s} } } \biggr\vert \biggr], \end{aligned}$$

where

$$\begin{aligned}& {M_{1}} = \mathop{\operatorname{Sup}}_{{m_{1}} < s < {n_{1}}} \bigl\vert {{f ^{{\diamondsuit _{{\alpha _{1}}}}}} \bigl( {s,{\sigma _{2}} ( t )} \bigr)} \bigr\vert , \qquad {M_{2}} =\mathop{\operatorname{Sup}} _{{m_{2}} < t < {n_{2}}} \bigl\vert {{f^{{\diamondsuit _{{\alpha _{2}}}}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr)} \bigr\vert \quad \mbox{and} \\& {M_{3}} = \mathop{\operatorname{Sup}} _{{m_{1}} < s < {n_{1}}, {m_{2}} < t < {n_{2}}} \bigl\vert {{f^{{\diamondsuit _{{\alpha _{1}}}}{\diamondsuit _{{\alpha _{2}}}}}} ( {s,t} )} \bigr\vert , \end{aligned}$$

which gives

$$\begin{aligned} & \biggl\vert {f ( {x,y} ) - \frac{1}{k} \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl( {{\sigma _{1}} ( s ),{\sigma _{2}} ( t )} \bigr){\diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s} } } \biggr\vert \\ &\quad \le \frac{1}{k} \biggl[ {{{M_{1}}} ( {{n_{2}} - {m_{2}}} )} \int _{{m_{1}}}^{{n_{1}}} {p ( {x,s} )} {\diamondsuit _{{\alpha _{1}}}}s + {M_{2}} ( {{n_{1}} - {m_{1}}} ) \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )} {\diamondsuit _{{\alpha _{2}}}}t \\ &\qquad {} + {M_{3}} \int _{{m_{1}}}^{{n_{1}}} {p ( {x,s} )} {\diamondsuit _{{\alpha _{1}}}}s \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )} {\diamondsuit _{{\alpha _{2}}}}t \biggr] \\ &\quad = \frac{1}{{ ( {{n_{1}} - {m_{1}}} )}}{M_{1}} \bigl[ {\hat{{h_{2}}} ( {x,{m_{1}}} ) - \hat{{h_{2}}} ( {x,{n_{1}}} )} \bigr] + \frac{1}{{ ( {{n_{2}} - {m_{2}}} )}}{M_{2}} \bigl[ { \hat{{h_{2}}} ( {y,{m_{2}}} ) - \hat{{h_{2}}} ( {y,{n_{2}}} )} \bigr] \\ & \qquad {}+ \frac{1}{k}{M_{3}} \bigl[ { \hat{{h_{2}}} ( {x,{m_{1}}} ) - \hat{{h_{2}}} ( {x,{n_{1}}} )} \bigr] \bigl[ { \hat{{h_{2}}} ( {y,{m_{2}}} ) - \hat{{h_{2}}} ( {y,{n_{2}}} )} \bigr], \end{aligned}$$

which is the required result. □