- Research
- Open access
- Published:
Some new local fractional inequalities associated with generalized \((s,m)\)-convex functions and applications
Advances in Difference Equations volume 2020, Article number: 406 (2020)
Abstract
Fractal analysis is one of interesting research areas of computer science and engineering, which depicts a precise description of phenomena in modeling. Visual beauty and self-similarity has made it an attractive field of research. The fractal sets are the effective tools to describe the accuracy of the inequalities for convex functions. In this paper, we employ linear fractals \(\mathbb{R}^{\alpha }\) to investigate the \((s,m)\)-convexity and relate them to derive generalized Hermite–Hadamard (HH) type inequalities and several other associated variants depending on an auxiliary result. Under this novel approach, we aim at establishing an analog with the help of local fractional integration. Meanwhile, we establish generalized Simpson-type inequalities for \((s,m)\)-convex functions. The results in the frame of local fractional showed that among all comparisons, we can only see the correlation between novel strategies and the earlier consequences in generalized s-convex, generalized m-convex, and generalized convex functions. We obtain application in probability density functions and generalized special means to confirm the relevance and computational effectiveness of the considered method. Similar results in this dynamic field can also be widely applied to other types of fractals and explored similarly to what has been done in this paper.
1 Introduction and prelimnaries
Fractional calculus based on differential and difference equations is of considerable importance due to their connection with real-world problems that depend not only on the instant time but also on the previous time, in particular, modeling the phenomena by means of fractals, random walk processes, control theory, signal processing, acoustics, and so on (see [1–12]). It has been shown that fractional-order models are much more adequate than integer-order models. A number of methods used to solve nonlinear partial differential equations have been successfully generalized to fractional differential equations, such as the Adomian decomposition method, homotopy analysis method, variational iteration method, transform method, symmetry group method, and invariant subspace method. The concepts of fractional differentiation and fractional integration were examined by Riemann, Liouville, Abel, Laurent, Hardy, and Littlewood. Detailed discussions of fractional calculus and related work can be found in [13–16]. Fractal analysis is an entirely new field of research based on fractional calculus. It has introduced some fascinating complex graphs, picture compressions, and computer graphics. In 1982, Benoit Mandelbrot [17], the father of fractal geometry, in his book “The Fractal Geometry of Nature” predicted that “clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” Individuals accept that the items in nature can be made or can be depicted by images, for example, lines, circles, conic areas, polygons, circles, and quadratic surfaces. The utilization of new scientific tools and concepts in this field of research will have an inordinate impression on enlightening image compression, where fractals and fractal-concerned techniques have demonstrated applications [18–20]. It is interesting that the authors [21, 22] investigated the local fractional functions on fractal space deliberately, which comprises of local fractional calculus and the monotonicity of functions. Numerous analysts contemplated the characteristics of functions on fractal space and built numerous sorts of fractional calculus by utilizing various strategies [23–25].
The connection among fractal sets, integral inequalities, and convexity is very strong. Therefore it is essential to create mathematical inequalities that inspect the fractal sets and their significance in various areas of mathematics and engineering problems. Convexity is utilized to portray the functional values of a framework that we normally deal with inequalities. Convex functions are firmly identified with the most celebrated HH inequality [26, 27], which is the principal essential consequence for convex functions with natural geometrical interpretation and numerous applications. It has attained considerably much interest in elementary mathematics and is stated as follows:
provided that \(\mathcal{H}:\mathcal{I}\subseteq \mathbb{R}\rightarrow \mathbb{R}\) is a convex function on an interval \(\mathcal{I}\) of reals with \(\lambda _{1},\lambda _{2}\in \mathcal{I}\) defined by
for \(\lambda _{1},\lambda _{2}\in \mathcal{I}\) and \(\zeta \in [0,1]\). For a concave function \(\mathcal{H}\), the inequalities in (1.1) hold in reverse direction. Over the last two decades, these types of generalizations have led to many novel testimonies, stimulating extensions, conspicuous generalizations, innovative HH-type inequalities, and a lot of applications of inequalities (1.1) in the literature of mathematical inequalities and in other branches of pure and applied mathematics; see [28–30] and the references therein.
Simpson’s inequality is widely studied in the literature occupying a significant place in numerical analysis and inequality theory due to its systematic nature and is stated as follows:
where the mapping \(\mathcal{H}:[\lambda _{1},\lambda _{2}]\rightarrow \mathbb{R}\) is assumed to be four times continuously differentiable on the interval, and \(\mathcal{H}^{4}\) to be bounded on \((\lambda _{1},\lambda _{2})\), that is, \(\|\mathcal{H}^{4}\|_{\infty }= \sup_{\zeta \in (\lambda _{1}, \lambda _{2})}\vert \mathcal{H}^{4}(\zeta )\vert <\infty \). For more generalizations of Simpson-type inequalities, see [31–36]. Adopting the aforementioned trend, we intend to establish some novel results on \((s,m)\)-convex functions via local fractional integrals. To be more precise, Simpson-type inequalities in local fractional integrals are also presented.
Toader [37] gave the idea of m-convexity as follows.
Definition 1.1
([37])
A mapping \(\mathcal{H}:[0,\mathfrak{b}^{*}]\rightarrow \mathbb{R}\) is known to be m-convex if
for \(m\in [0,1]\), \(\lambda _{1},\lambda _{2}\in [0,\mathfrak{b}^{*}]\), and \(\zeta \in [0,1]\).
Also, \(\mathcal{K}_{m}(\mathfrak{b})\) denotes the set of m-convex functions on \([\lambda _{1},\lambda _{2}]\) for which \(\mathcal{H}(0)\leq 0\). For some modifications and generalizations related to m-convex functions, we refer to [28, 38].
Hudzik and Maligranda [39] proposed, among others, a class of functions, known as s-convex functions, defined as follows.
Definition 1.2
([39])
A mapping \(\mathcal{H}:\mathcal{I}\subseteq \mathbb{R}\rightarrow \mathbb{R}\) is said be s-convex if
for all \(\lambda _{1},\lambda _{2}\in \mathcal{I},\zeta \in [0,1]\) and some fixed \(s\in (0,1]\).
Clearly, we see that for \(s=1\), s-convexity becomes the classical convexity of functions on \([0,\infty )\). For generalizations and refinements on s-convex and m-convex functions, we refer the readers to [28, 38–42].
Now we mention some preliminaries from the theory of local fractional calculus. These ideas and important consequences associated with the local fractional derivative and local fractional integral are mainly due to Yang [22].
Let \(\nu _{1}^{\alpha ^{*}},\nu _{2}^{\alpha ^{*}}\), and \(\nu _{3}^{\alpha ^{*}}\) belong to \(\mathbb{R}^{\alpha ^{*}} (0<\alpha ^{*}\leq 1)\). Then
-
(1)
\(\nu _{1}^{\alpha ^{*}}+\nu _{2}^{\alpha ^{*}}\) and \(\nu _{1}^{\alpha ^{*}}\nu _{2}^{\alpha ^{*}}\) belong to \(\mathbb{R}^{\alpha ^{*}}\);
-
(2)
\(\nu _{1}^{\alpha ^{*}}+\nu _{2}^{\alpha ^{*}}=\nu _{2}^{\alpha ^{*}}+ \nu _{1}^{\alpha ^{*}}=(\nu _{1}+\nu _{2})^{\alpha ^{*}}=(\nu _{2}+ \nu _{1})^{\alpha ^{*}}\);
-
(3)
\(\nu _{1}^{\alpha ^{*}}+(\nu _{2}^{\alpha ^{*}}+\nu _{3}^{\alpha ^{*}})=( \nu _{1}^{\alpha ^{*}}+\nu _{2}^{\alpha ^{*}})+\nu _{3}^{\alpha ^{*}}\);
-
(4)
\(\nu _{1}^{\alpha ^{*}}\nu _{2}^{\alpha ^{*}}=\nu _{2}^{\alpha ^{*}} \nu _{1}^{\alpha ^{*}}=(\nu _{1}\nu _{2})^{\alpha ^{*}}=(\nu _{2}\nu _{1})^{ \alpha ^{*}}\);
-
(5)
\(\nu _{1}^{\alpha ^{*}}(\nu _{2}^{\alpha ^{*}}\nu _{3}^{\alpha ^{*}})=( \nu _{1}^{\alpha ^{*}}\nu _{2}^{\alpha ^{*}})\nu _{3}^{\alpha ^{*}}\);
-
(6)
\(\nu _{1}^{\alpha ^{*}}(\nu _{2}^{\alpha ^{*}}+\nu _{3}^{\alpha ^{*}})= \nu _{1}^{\alpha ^{*}}\nu _{2}^{\alpha ^{*}}+\nu _{1}^{\alpha ^{*}} \nu _{3}^{\alpha ^{*}}\);
-
(7)
\(\nu _{1}^{\alpha ^{*}}+0^{\alpha ^{*}}=0^{\alpha ^{*}}+\nu _{1}^{ \alpha ^{*}}=\nu _{1}^{\alpha ^{*}}\) and \(\nu _{1}^{\alpha ^{*}}1^{\alpha ^{*}}=1^{\alpha ^{*}}\nu _{1}^{ \alpha ^{*}}=\nu _{1}^{\alpha ^{*}}\).
Definition 1.3
A nondifferentiable mapping \(\mathcal{H}:\mathbb{R}\rightarrow \mathbb{R}^{\alpha ^{*}}, \theta \rightarrow \mathcal{H}(\epsilon )\), is said to be local fractional continuous at \(\epsilon _{\circ }\) if for any \(\epsilon >0\), there exists \(l>0\) such that
for \(\vert \epsilon -\epsilon _{\circ }\vert <\kappa \). If \(\mathcal{H}(\epsilon )\) is local continuous on \((\lambda {1},\lambda {2})\), then we write \(\mathcal{H}(\epsilon )\in \mathbb{C}_{\alpha ^{*}}(\lambda {1}, \lambda {2})\).
Definition 1.4
The local fractional derivative of \(\mathcal{H}(\epsilon )\) of order \(\alpha ^{*}\) at \(\epsilon =\epsilon _{\circ }\) is defined by
where \(\Delta ^{\alpha ^{*}}(\mathcal{H}(\epsilon )-\mathcal{H}(\epsilon _{ \circ }))=\varGamma (\alpha ^{*}+1)(\mathcal{H}(\epsilon )-\mathcal{H}( \epsilon _{\circ }))\). Let \(\mathcal{H}^{(\alpha ^{*})}(\epsilon )=\mathcal{D}_{\epsilon }^{ \alpha ^{*}}\mathcal{H}(\epsilon )\). If there exists \(\mathcal{H}^{(k+1)\alpha ^{*}}(\epsilon )= \overbrace{\mathcal{D}_{\epsilon }^{\alpha ^{*}}...\mathcal{D}_{\epsilon }^{\alpha ^{*}}}^{(k+1) times}\mathcal{H}(\epsilon )\) for any \(\epsilon \in \varOmega \subseteq \mathbb{R}\), then we write \(\mathcal{H}\in \mathcal{D}_{(k+1)\alpha ^{*}}(\mathcal{I})\), where \(k=0,1,2,\ldots\) .
Definition 1.5
Let \(\mathcal{H}(\epsilon )\in \mathbb{C}_{\alpha ^{*}}[\lambda {1}, \lambda {2}]\), and let \(\Delta =\{\eta _{0},\eta _{1},\ldots,\eta _{N}\}\ (N\in \mathbb{N})\) be a partition of \([\lambda {1},\lambda {2}]\) such that \(\lambda {1}=\eta _{0}<\eta _{1}<\cdots<\eta _{N}=\lambda {2}\). Then the local fractional integral of \(\mathcal{H}\) on \([\lambda {1},\lambda {2}]\) of order \(\alpha ^{*}\) is defined as follows:
where \(\delta \eta:=\max \{\Delta \eta _{1},\Delta \eta _{2},\ldots,\Delta \eta _{N-1}\}\) and \(\Delta \eta _{j}:=\eta _{j+1}-\eta _{j}, j=0,\ldots,N-1\).
It follows that \(_{\lambda {1}}\mathcal{I}_{\lambda {2}}^{(\alpha ^{*})}\mathcal{H}( \epsilon )=0\) if \(\lambda {1}=\lambda {2}\) and \(_{\lambda {1}}\mathcal{I}_{\lambda {2}}^{(\alpha ^{*})}\mathcal{H}( \epsilon )= - _{\lambda {2}}\mathcal{I}_{\lambda {1}}^{(\alpha ^{*})} \mathcal{H}(\epsilon )\) if \(\lambda {1}<\lambda {2}\). For any \(\epsilon \in [\lambda {1},\lambda {2}]\), if there exists \(_{\lambda {1}}\mathcal{I}_{\lambda {2}}^{(\alpha ^{*})}\mathcal{H}( \epsilon )\), then we write \(\mathcal{H}(\epsilon )\in \mathcal{I}_{\epsilon }^{\alpha ^{*}}[ \lambda {1},\lambda {2}]\).
Lemma 1.6
([22])
-
(1)
Suppose that\(\mathcal{H}(u)=\mathcal{G}^{(\alpha ^{*})}(u)\in \mathbb{C}_{\alpha ^{*}}[ \lambda {1},\lambda {2}]\). Then
$$\begin{aligned} _{\lambda {1}}\mathcal{I}_{\lambda {2}}^{(\alpha ^{*})}\mathcal{H}(u)= \mathcal{G}(\lambda {2})-\mathcal{G}(\lambda {1}). \end{aligned}$$ -
(2)
Suppose that\(\mathcal{H}(u), \mathcal{G}(u)\in \mathcal{D}_{\alpha ^{*}}[\lambda {1}, \lambda {2}]\)and\(\mathcal{H}^{(\alpha ^{*})}(u),\mathcal{G}^{(\alpha ^{*})}(u)\in \mathbb{C}_{\alpha ^{*}}[\lambda {1},\lambda {2}]\). Then
$$\begin{aligned} _{\lambda {1}}\mathcal{I}_{\lambda {2}}^{(\alpha ^{*})}\mathcal{H}(u) \mathcal{G}^{(\alpha ^{*})}(u)=\mathcal{H}(u)\mathcal{G}(u)\vert _{ \lambda {1}}^{\lambda {2}}-{}_{\lambda {1}}\mathcal{I}_{\lambda {2}}^{( \alpha ^{*})} \mathcal{H}^{(\alpha ^{*})}(u)\mathcal{G}(u). \end{aligned}$$
Lemma 1.7
([22])
Lemma 1.8
([43] Generalized Hölder’s inequality)
Let\(s,q>1\)with\(s^{-1}+q^{-1}=1\), and let\(\mathcal{H},\mathcal{G}\in \mathbb{C}_{\alpha ^{*}}[\lambda {1}, \lambda {2}]\). Then
Mo et al. [24] derived the following generalized HH inequality for generalized s-convex functions:
In 1994, Hudzik and Maligranda [39] provided several generalizations via s-convexity and presented intriguing outcomes about the HH inequality for s-convex functions. In 1915, Bernstein and Doetsch [40] established a variant of the HH inequality for s-convex functions in the second sense. Moreover, some well-known integral inequalities via local fractional integral have been studied by several researchers; for instance, Kilicman and Saleh [41, 42] derived generalized HH inequalities for generalized s-convex functions. Du et al. [38] contemplated certain inequalities for generalized m-convex functions on fractal sets with utilities. Also, Vivas et al. [44] explored generalized Jensen and HH inequalities for h-convex functions. For results associated with local fractional inequalities, we refer the interested readers to [24, 45–47] and the references therein.
Owing to the phenomena mentioned, the principal purpose of this research is exploring a novel concept of \((s,m)\)-convex functions, and we address important properties for such functions. Also, we establish some novel variants, which interact between \((s,m)\)-convex functions and local fractional integrals. In fractal sets, we carry out two novel generalized identities to investigate the local differentiability of \((s,m)\)-convex functions, s-convex functions, and generalized m-convex functions. Meanwhile, we present some new generalized Simpson-type inequalities for \((s,m)\)-convexity. Generalized new special cases show the impressive performance of the local fractional integration. Some special cases correlate with existing results in classical convexity.
2 Generalized \((s,m)\)-convex functions
We now present the concept of generalized \((s,m)\)-convex functions on a fractal space.
Definition 2.1
Let \(s\in (0,1]\). A function \(\mathcal{H}:[0,\mathfrak{b}^{*}]\rightarrow \mathbb{R}^{\alpha ^{*}}\) with \(\mathfrak{b}^{*}>0\) is said to be generalized \((s,m)\)-convex if
for \(u,v\in [0,\mathfrak{b}^{*}],s\in (0,1]\), and some fixed \(m\in [0,1]\).
Remark 2.2
Definition 2.1 leads to the conclusion that
-
1.
If we take \(s=1\), then we get Definition 2.1 in [38].
-
2.
If we take \(s=1\) and \(\alpha ^{*}=1\), then we get Definition in [39].
-
3.
If we take \(m=1\), then we get Definition in [23].
-
4.
If we take \(m=1\) and \(\alpha ^{*}=1\), then we get Definition in [37].
-
5.
If we take \(m=1\) and \(s=1\), then we get Definition in [24].
-
6.
If we take \(m=s=1\) and \(\alpha ^{*}=1\), then we get the concept of classical convex functions.
Moreover, if we take \(\zeta =\frac{1}{2}\) in (2.1), then the generalized \((s,m)\)-convex functions become Jensen-type generalized \((s,m)\)-convex functions as follows:
for \(u,v\in [0,\mathfrak{b}^{*}],s\in (0,1]\), and for some fixed \(m\in [0,1]\).
It is worth mentioning that \((s,m)\)-convex functions reduce to generalized convex, generalized m-convex functions, and generalized s-convex functions as particular cases. This shows that outcomes derived in the present paper continue to hold for these classes of convex functions and their variant forms.
Proposition 2.3
For\(m\in [0,1]\)and\(s\in (0,1]\), if\(\mathcal{H},\mathcal{G}:\varOmega \rightarrow \mathbb{R}^{\alpha ^{*}}\)is a generalized\((s,m)\)-convex functions, then
-
(1)
\(\mathcal{H}+\mathcal{G}\)is a generalized\((s,m)\)-convex function;
-
(2)
\(\lambda ^{\alpha ^{*}} \mathcal{H}\)is a generalized\((s,m)\)-convex function.
Proof
(1) Since \(\mathcal{H}\) and \(\mathcal{G}\) are generalized \((s,m)\)-convex functions on Ω and \(\zeta \in [0,1]\), we have
So, \(\mathcal{H}+\mathcal{G}\) is a generalized \((s,m)\)-convex function on Ω.
(2) Since \(\mathcal{H}\) and \(\mathcal{G}\) are generalized \((s,m)\)-convex functions on Ω, for \(\zeta \in [0,1]\) and \(\lambda \in \mathbb{R}_{+}\), we have
and hence \(\lambda ^{\alpha ^{*}}\mathcal{H}\) is a generalized \((s,m)\)-convex function on Ω. □
Proposition 2.4
Let\(\mathcal{H}_{n}:\varOmega \rightarrow \mathbb{R}^{\alpha ^{*}}\), \(n\in \mathbb{N}\), be a sequence of generalized\((s,m)\)-convex functions converging pointwise to a function\(\mathcal{H}:\varOmega \rightarrow \mathbb{R}^{\alpha ^{*}}\). Then\(\mathcal{H}\)is a generalized\((s,m)\)-convex function onΩ.
Proof
Let \(u,v\in \varOmega, \zeta \in [0,1]\), and let \(\lim_{n \rightarrow \infty }\mathcal{H}_{n}(u)=\mathcal{H}(u)\). Then
that is, \(\mathcal{H}\) is a generalized \((s,m)\)-convex function on Ω. □
Proposition 2.5
For\(m\in [0,1]\)and\(s\in (0,1]\), let\(\mathcal{H}:[0,\infty )\rightarrow \mathbb{R}^{\alpha ^{*}}\)be a generalized\((s,m)\)-convex function such that\(0\leq \lambda _{1}< m\lambda _{2}<\infty \). If\(\mathcal{H}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1},m\lambda _{2}]\), then
Proof
Utilizing the generalized \((s,m)\)-convexity of \(\mathcal{H}\), for all \(\zeta \in [0,1]\) and \(\lambda _{1},\lambda _{2}\in \varOmega \), we have
and
Adding these inequalities, we get
Integrating inequality (2.4) with respect to ζ over \((0,1)\), we have
Therefore we get
the desired result. □
3 Certain new results on generalized \((s, m)\)-convexity
This section is devoted to the generalized HH inequality for generalized \((s,m)\)-convex functions via local fractional integrals.
Theorem 3.1
For\(s, m\in (0,1]\), let\(\mathcal{H}:\varOmega \rightarrow \mathbb{R}^{\alpha ^{*}}\)be a generalized\((s,m)\)-convex function defined on a fractal space. If\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1},m \lambda _{2}]\)for some\(0\leq \lambda _{1}<\lambda _{2}\), then
Proof
Taking into account inequality (2.2), for all \(u,v\in \varOmega \), we have
Substituting \(u=\frac{\zeta }{2}\lambda _{1}+m\frac{2-\zeta }{2}\lambda _{2}, v= \frac{2-\zeta }{2m}\lambda _{1}+\frac{\zeta }{2}\lambda _{2}\), for all \(\zeta \in [0,1]\), we have
Integrating this inequality with respect to ζ over \((0,1)\), we have
Also, using the fact that
we have
For the proof of the second inequality in (3.1), noting that \(\mathcal{H}\) is a generalized \((s,m)\)-convex function, for \(\zeta \in [0,1]\), we have
Integrating this inequality with respect to ζ over \((0,1)\), we have
where we have used Lemma 1.7 and the fact that
This completes the proof. □
We present some remarkable cases of Theorem 3.1 as corollaries and remarks.
I. If we take \(s=1\), then we have a new result for generalized m-convex functions.
Corollary 3.2
For\(m\in (0,1]\), let\(\mathcal{H}:\varOmega \rightarrow \mathbb{R}^{\alpha ^{*}}\)be a generalizedm-convex function defined on a fractal space. If\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1},m \lambda _{2}]\)for some\(0\leq \lambda _{1}<\lambda _{2}\), then
II. If we take \(m=1\), then we have a new result for s-convex functions.
Corollary 3.3
For\(s\in (0,1]\), let\(\mathcal{H}:\varOmega \rightarrow \mathbb{R}^{\alpha ^{*}}\)be a generalizeds-convex function defined on a fractal space. If\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1}, \lambda _{2}]\)for some\(0\leq \lambda _{1}<\lambda _{2}\), then
III. If we take \(m=1\) and \(s=1\), then we have a new result for generalized convex functions.
Corollary 3.4
For\(s\in (0,1]\), let\(\mathcal{H}:\varOmega \rightarrow \mathbb{R}^{\alpha ^{*}}\)be a generalized convex function defined on a fractal space. If\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1}, \lambda _{2}]\)for some\(0\leq \lambda _{1}<\lambda _{2}\), then
Remark 3.5
If we choose \(\alpha ^{*}=1\) and \(m=s=1\), then Theorem 3.1 reduces to inequality (1.1).
Further, we obtain novel bounds that refine the generalized HH inequality for functions whose first derivative in absolute value raised to a certain power greater than one, respectively, at least one, is a generalized \((s,m)\)-convex function. For our further results, we need the following lemma.
Lemma 3.6
For\(m\in (0,1]\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\) (\(\varOmega ^{\circ }\)is the interior ofΩ) be a function such that\(\mathcal{H}\in \mathcal{D}_{\alpha ^{*}}(\varOmega ^{\circ })\)and\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1},m \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). Then
Proof
Utilizing local fractional integration by parts, we get
Analogously, we have
Adding \(I_{1}\) and \(I_{2}\), we get the desired result. □
Theorem 3.7
For\(s, m\in (0,1]\)and\(p, q > 1\)with\(p^{-1}+q^{-1}=1\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\)be a differentiable function on\(\varOmega ^{\circ }\)such that\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1},m \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). If\(\vert \mathcal{H}^{(\alpha ^{*})}\vert ^{q}\)is generalized\((s,m)\)-convex onΩfor\(q> 1\), then
Proof
Using Lemma 3.6, the generalized power mean inequality, and the generalized \((s,m)\)-convexity of \(\vert \mathcal{H}^{(\alpha ^{*})}\vert ^{q}\), we have
Taking into consideration Lemma 1.7, we easily see that
Combining (3.8)–(3.10), we get the desired inequality (3.7). This completes the proof. □
Some particular cases of Theorem 3.7 are presented as follows.
I. If we choose \(s=1\), then we get a new result for generalized m-convex functions.
Corollary 3.8
For\(m\in (0,1]\)and\(p, q > 1\)with\(p^{-1}+q^{-1}=1\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\)be a differentiable function on\(\varOmega ^{\circ }\)such that\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1},m \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). If\(\vert \mathcal{H}^{(\alpha ^{*})}\vert ^{q}\)is generalizedm-convex onΩfor\(q> 1\), then
II. If we choose \(m=1\), then we get a new result for generalized s-convex functions.
Corollary 3.9
For\(s\in (0,1]\)and\(p, q > 1\)with\(p^{-1}+q^{-1}=1\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\)be a differentiable function on\(\varOmega ^{\circ }\)such that\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1}, \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). If\(\vert \mathcal{H}^{(\alpha ^{*})}\vert ^{q}\)is generalizeds-convex onΩfor\(q> 1\), then
III. If we choose \(m=1\) and \(s=1\), then we get a new result for generalized convex functions.
Corollary 3.10
For\(p, q > 1\)with\(p^{-1}+q^{-1}=1\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\)be a differentiable function on\(\varOmega ^{\circ }\)such that\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1}, \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). If\(\vert \mathcal{H}^{(\alpha ^{*})}\vert ^{q}\)is generalized convex onΩfor\(q>1\), then
Remark 3.11
If we choose \(\alpha ^{*}=1\) and \(s=m=1\), then Theorem 3.7 reduces to the result in [48].
Theorem 3.12
For\(s, m\in (0,1]\)and\(p, q > 1\)with\(p^{-1}+q^{-1}=1\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\)be a differentiable function on\(\varOmega ^{\circ }\)such that\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1},m \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). If\(\vert \mathcal{H}^{(\alpha ^{*})}\vert ^{q}\)is generalized\((s,m)\)-convex onΩfor\(q> 1\), then
Proof
Using Lemma 3.6, the generalized Hölder inequality, and the generalized \((s,m)\)-convexity of \(\vert \mathcal{H}^{(\alpha ^{*})}\vert ^{q}\), we have
the required result. This completes the proof. □
We present some particular cases of Theorem 3.12.
I. If we take \(s=1\), then we get a new result for generalized m-convex functions.
Corollary 3.13
For\(m\in (0,1]\)and\(p, q > 1\)with\(p^{-1}+q^{-1}=1\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\)be a differentiable function on\(\varOmega ^{\circ }\)such that\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1},m \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). If\(\vert \mathcal{H}^{(\alpha ^{*})}\vert ^{q}\)is generalizedm-convex onΩfor\(q> 1\), then
II. If we take \(m=1\), then we get a new result for generalized s-convex functions.
Corollary 3.14
For\(s\in (0,1]\)and\(p, q > 1\)with\(p^{-1}+q^{-1}=1\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\)be a differentiable function on\(\varOmega ^{\circ }\)such that\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1}, \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). If\(\vert \mathcal{H}^{(\alpha ^{*})}\vert ^{q}\)is generalizeds-convex onΩfor\(q> 1\), then
III. If we take \(m=1=s\), then we get a new result for generalized convex functions.
Corollary 3.15
For\(p, q > 1\)with\(p^{-1}+q^{-1}=1\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\)be a differentiable function on\(\varOmega ^{\circ }\)such that\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1}, \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). If\(\vert \mathcal{H}^{(\alpha ^{*})}\vert ^{q}\)is generalized convex onΩfor\(q> 1\), then
Remark 3.16
If we choose \(\alpha ^{*}=1\) and \(s=m=1\), then Theorem 3.12 reduces to the result in [48].
4 New estimates for generalized Simpson’s type via \((s,m)\)-convex functions
Before continuing toward our main results in regards to generalized Simpson’s inequality utilizing m-convex functions, we start with the accompanying lemma.
Lemma 4.1
For\(m\in (0,1]\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\) (\(\varOmega ^{\circ }\)is the interior ofΩ) be such that\(\mathcal{H}\in \mathcal{D}_{\alpha ^{*}}(\varOmega ^{\circ })\)and\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1},m \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). Then
Proof
Consider
Using the local fractional integration by parts, we get
and
Combining (2.1), (4.3), and Definition 2.1, suitable rearrangements give the desired identity (4.1). □
Theorem 4.2
For\(m\in (0,1]\)and\(s\in (0,1]\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\)be a differentiable function on\(\varOmega ^{\circ }\)such that\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1},m \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). If\(\vert \mathcal{H}^{(\alpha ^{*})}\vert \)is generalized\((s,m)\)-convex onΩ, then
Proof
Utilizing Lemma 4.1, the modulus property, and generalized \((s,m)\)-convexity of \(\vert \mathcal{H}^{\alpha ^{*}}\vert \), we obtain
Utilizing Lemma 1.7, we get
and
Again, using Lemma 1.7 and change of variable \(u=1-\zeta \), we have
and, similarly,
and
Combining (4.6) and (4.8)–(4.15), we get the desired inequality (4.5). This completes the proof. □
We present aome particular cases of Theorem 4.2.
I. If we choose \(s=1\), then we get a new result for generalized m-convex functions.
Corollary 4.3
For\(m\in (0,1]\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\)be a differentiable function on \(\varOmega ^{\circ }\)such that\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1},m \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). If\(\vert \mathcal{H}^{(\alpha ^{*})}\vert \)is generalizedm-convex onΩ, then
II. If we choose \(m=1\), then we get a new result for generalized s-convex functions.
Corollary 4.4
For\(s\in (0,1]\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\)be a differentiable function on\(\varOmega ^{\circ }\)such that\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1}, \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). If\(\vert \mathcal{H}^{(\alpha ^{*})}\vert \)is generalizeds-convex onΩ, then
Remark 4.5
If we choose \(s=1=m\), then Theorem 4.2 reduces to Theorem 7 in [36].
Theorem 4.6
For\(m,s \in (0,1]\)and\(p,q > 1\)with\(p^{-1}+q^{-1}=1\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\)be a differentiable function on\(\varOmega ^{\circ }\)such that\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1},m \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). If\(\vert \mathcal{H}^{(\alpha ^{*})}\vert ^{q}\)is generalized\((s,m)\)-convex onΩfor\(q> 1\), then
Proof
Utilizing Lemma 4.1, the modulus property, and generalized \((s,m)\)-convexity of \(\vert \mathcal{H}^{\alpha ^{*}}\vert \), we obtain
Using Lemma 1.7, and change of variable technique, we obtain
□
I. If we choose \(s=1\), then we get a new result for generalized m-convex functions.
Corollary 4.7
For\(m\in (0,1]\)and\(p, q > 1\)with\(p^{-1}+q^{-1}=1\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\)be a differentiable function on\(\varOmega ^{\circ }\)such that\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1},m \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). If\(\vert \mathcal{H}^{(\alpha ^{*})}\vert ^{q}\)is generalizedm-convex onΩfor\(q> 1\), then
II. If we choose \(m=1\), then we get a new result for generalized s-convex functions.
Corollary 4.8
For\(s\in (0,1]\)and\(p, q > 1\)with\(p^{-1}+q^{-1}=1\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\)be a differentiable function on\(\varOmega ^{\circ }\)such that\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1},m \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). If\(\vert \mathcal{H}^{(\alpha ^{*})}\vert ^{q}\)is generalizeds-convex onΩfor\(q> 1\), then
III. If we choose \(m=1=s\), then we get a new result for generalized convex functions.
Corollary 4.9
For\(p, q > 1\)with\(p^{-1}+q^{-1}=1\), let\(\mathcal{H}:\varOmega ^{\circ }\rightarrow \mathbb{R}^{\alpha ^{*}}\)be a differentiable function on\(\varOmega ^{\circ }\)such that\(\mathcal{H}^{(\alpha ^{*})}\in \mathbb{C}_{\alpha ^{*}}[\lambda _{1}, \lambda _{2}]\)for\(\lambda _{1},\lambda _{2}\in \varOmega ^{\circ }\)with\(\lambda _{2}>\lambda _{1}\). If\(\vert \mathcal{H}^{(\alpha ^{*})}\vert ^{q}\)is generalized convex onΩfor\(q> 1\), then
5 Applications
5.1 Probability density functions
Consider a random variable χ with generalized probability density function \(\mathfrak{p}:[\lambda _{1},\lambda _{2}]\rightarrow [0^{\alpha ^{*}},1^{ \alpha ^{*}}]\), which is generalized convex and has the cumulative distribution function
Moreover, the generalized expectation can be expressed as
For more information related to probability density functions, see [49].
Clearly, we see that
The following results are associated with Sect. 4.
Proposition 5.1
In Theorem4.2, choosing\(m=1=s\), we have
Proposition 5.2
In Theorem4.6, choosing\(m=1=s\), we have
5.2 Generalized special means
Considering the following \(\alpha ^{*}\)-type special means [50]. For \(\lambda _{1}<\lambda _{2}\) and \(\lambda _{1},\lambda _{2}\in \mathbb{R}^{\alpha ^{*}}\), we have:
I. The generalized arithmetic mean
II. The generalized logarithmic mean
Considering \(\mathcal{H}(u)=u^{n\alpha ^{*}} (u\in \mathbb{R}:n\in \mathbb{Z},\vert n\vert \geq 2)\) in Corollaries 3.10 and 3.15, we obtain the following inequalities stated by Propositions 5.3 and 5.4, respectively.
Proposition 5.3
Let\(\lambda _{1},\lambda _{2}{\in \mathbb{R}}\)with\(\lambda _{1}<\lambda _{2}, 0\notin [\lambda _{1},\lambda _{2}]\), and\(n\in \mathbb{N}\setminus \{1\}\). Then
Proposition 5.4
Let\(\lambda _{1},\lambda _{2}{\in \mathbb{R}}\)with\(\lambda _{1}<\lambda _{2}, 0\notin [\lambda _{1},\lambda _{2}]\), and\(n\in \mathbb{N}\setminus \{1\}\). Then
6 Conclusions
In this paper, we addressed a novel concept of \((s,m)\)-convex functions on a fractal domain. Moreover, we have discussed some algebraic properties of the proposed technique. Also, we established some appropriate results about generalized HH type inequalities and local fractional Simpson’s-like type inequalities by using tools of fractal analysis and \((s,m)\)-convexity. Several novel results have been captured for generalized s-convex, generalized m-convex, and generalized convex functions. The obtained results have been testified by two intriguing applications to show the effectiveness of the derived results. To the best of our knowledge, the said results are new for convexity theory involving fractal sets. In the future the above theory and analysis can be extended to more complicated and applicable problems of convexity involving fractal domains. Finally, our consequences have a potential connection in fractal theory and machine learning [19, 20]. This new concept will be opening new doors of investigation toward fractal differentiations and integrations in convexity, preinvexity, fractal image processing, and camouflage in the garments industry. We hope that the main results of this paper will inspire the interested readers and will stimulate further research in this field.
References
Atangana, A.: Modelling the spread of COVID-19 with new fractal-fractional operators: can the lockdown save mankind before vaccination? Chaos Solitons Fractals 136, 109860 (2020). https://doi.org/10.1016/j.chaos.2020.109860
Boccaletti, S., Ditto, W., Mindlin, G., Atangana, A.: Modeling and forecasting of epidemic spreading: the case of Covid-19 and beyond. Chaos Solitons Fractals 135, 109794 (2020). https://doi.org/10.1016/j.chaos.2020.109794
Kumar, S., Atangana, A.: A numerical study of the nonlinear fractional mathematical model of tumor cells in presence of chemotherapeutic treatment. Int. J. Biomath. 13(3), 2050021 (2020). https://doi.org/10.1142/S1793524520500217
Ghanbari, B., Atangana, A.: A new application of fractional Atangana–Baleanu derivatives: designing ABC-fractional masks in image processing. Physica A 542, 123516 (2020). https://doi.org/10.1016/j.physa.2019.123516
Danane, J., Allali, K., Hammouch, Z.: Mathematical analysis of a fractional differential model of HBV infection with antibody immune response. Chaos Solitons Fractals 136, 109787 (2020). https://doi.org/10.1016/j.chaos.2020.109787
Gao, W., Veeresha, P., Prakasha, D.G., Baskonus, H.M.: Novel dynamic structures of 2019-nCoV with nonlocal operator via powerful computational technique. Biology 9(5), 107 (2020). https://doi.org/10.3390/biology9050107
Khan, H., Tunç, C., Khan, A.: Green function’s properties and existence theorems for nonlinear singular-dalay-fractional differential equations. Discrete Contin. Dyn. Syst., Ser. S (2020). https://doi.org/10.3934/dcdss.2020139
Khan, H., Khan, A., Abdeljawad, T., Alkhazzan, A.: Existence results in Banach space for a nonlinear impulsive system. Adv. Differ. Equ. 2019, Article ID 18 (2019)
Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)
Abdeljawad, T., Baleanu, D.: Monotonicity results for fractional difference operators with discrete exponential kernels. Adv. Differ. Equ. 2017, Article ID 78 (2017)
Jarad, F., Ugurlu, E., Abdeljawad, T., Baleanu, D.: On a new class of fractional operators. Adv. Differ. Equ. 2018, 142 (2018)
Jarad, F., Abdeljawad, T.: Generalized fractional derivatives and Laplace transform. Discrete Contin. Dyn. Syst., Ser. S 13(3), 709–722 (2020)
Khan, H., Abdeljawad, T., Tunç, C., Alkhazzan, A., Khan, A.: Minkowski’s inequality for the AB-fractional integral operator. J. Inequal. Appl. 2019, Article ID 96 (2019). https://doi.org/10.1186/s13660-019-2045-3
Abdeljawad, T., Rashid, S., Khan, H., Chu, Y.-M.: On new fractional integral inequalities for p-convexity within interval-valued functions. Adv. Differ. Equ. 2020, 330 (2020). https://doi.org/10.1186/s13662-020-02782-y
Rashid, S., Akdemir, A.O., Niasr, K.S., Abdeljawad, T., Rahman, G.: New generalized reverse Minkowski and related integral inequalities involving generalized fractional conformable integrals. J. Inequal. Appl. 2020, 177 (2020). https://doi.org/10.1186/s13660-020-02445-2
Rashid, S., Abdeljawad, T., Jarad, F., Noor, M.A.: Some estimates for generalized Riemann–Liouville fractional integrals of exponentially convex functions and their applications. Mathematics 7(9), 807 (2019). https://doi.org/10.3390/math7090807
Mandelbrot, B.B.: The Fractal Geometry of Nature, vol. 2. WH Freeman, New York (1982)
Julia, G.: Mémoire sur l’itération des functions rationnelles. J. Math. Pures Appl. 8, 737–747 (1918)
Kwun, Y.C., Shahid, A.A., Nazeer, W., Abbas, M., Kang, S.M.: Fractal generation via CR iteration scheme with s-convexity. IEEE Access 7, 69986–69997 (2019)
Kumari, S., Kumari, M., Chugh, R.: Generation of new fractals via SP orbit with s-convexity. Int. J. Eng. Technol. 9(3), 2491–2504 (2017)
Yang, J., Baleanu, D., Yang, X.J.: Analysis of fractal wave equations by local fractional Fourier series method. Adv. Math. Phys. 2013, Article ID 632309 (2013)
Yang, X.J.: Advanced Local Fractional Calculus and Its Applications. World Science Publisher, New York (2012)
Mo, H.X., Sui, X.: Hermite–Hadamard-type inequalities for generalized s-convex functions on real linear fractal set \(\mathbb{R}^{\alpha ^{*}}(0\leq \alpha ^{*}<1)\). Math. Sci. 11, 241–246 (2017)
Mo, H.X., Sui, X., Yu, D.Y.: Generalized convex functions on fractal sets and two related inequalities. Abstr. Appl. Anal. 2014, 636751 (2014)
Mo, H.: Generalized Herrmite–Hadamard type inequalities involving local fractional integrals. Proc. Rom. Acad. 2014, 8 (2014)
Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier dune fonction considérée par Riemann. J. Math. Pures Appl. 58, 171–215 (1893)
Hermite, C.: Sur deux limites dune integrale definie. Mathesis 82, 3 (1883)
Rashid, S., Akdemir, A.O., Noor, M.A., Noor, K.I.: Generalization of inequalities analogous to preinvex functions via extended generalized Mittag-Leffler functions. In: Proceedings of the International Conference on Applied and Engineering Mathematics-Second International Conference, ICAEM 2018, 27–29 August Hitec Taxila, Pakistan (2018)
Rashid, S., Noor, M.A., Noor, K.I., Chu, Y.-M.: Ostrowski type inequalities in the sense of generalized K-fractional integral operator for exponentially convex functions. AIMS Math. 5(3), 2629–2645 (2020). https://doi.org/10.3934/math.2020171
Rashid, S., Jarad, F., Noor, M.A., Kalsoom, H., Chu, Y.-M.: Inequalities by means of generalized proportional fractional integral operators with respect to another function. Mathematics 1225, 7 (2020). https://doi.org/10.3390/math7121225
Alomari, M.W., Darus, M., Dragomir, S.S.: New inequalities of Simpson’s type for s-convex functions with applications. RGMIA Res. Rep. Collect. 12(4), Article ID 9 (2009)
Dragomir, S.S., Agarwal, R.P., Cerone, P.: On Simpson’s inequality and applications. J. Inequal. Appl. 5, 533–579 (2000)
Liu, B.Z.: An inequality of Simpson type. Proc. R. Soc. A 461, 2155–2158 (2005)
Park, J.: Generalization of some Simpson-like type inequalities via differentiable s-convex mappings in the second sense. Int. J. Math. Sci. 2011, Article ID 493531 (2011)
Rashid, S., Akdemir, A.O., Jarad, F., Noor, M.A., Noor, K.I.: Simpson’s type integral inequalities for K-fractional integrals and their applications. AIMS Math. 4(4), 1087–1100 (2019). https://doi.org/10.3934/math.2019.4.1087
Sarikaya, M.Z., Budak, H., Erden, S.: On new inequalities of Simpson’s type for generalized convex functions. Korean J. Math. 27(2), 279–295 (2019). https://doi.org/10.11568/kjm.2019.27.2.279
Toader, G.H.: Some generalizations of the convexity. In: Proceedings of Colloquium on Approximation and Optimixzation, Univ. Cluj. Napoca, Romania, pp. 329–338 (1984)
Du, T., Wang, H., Adil Khan, M., Zhang, Y.: Certain integral inequalities considering generalized m-convexity of fractals sets and their applications. Fractals 27(7), 17 (2019). https://doi.org/10.1142/So218348X19501172
Hudzik, H., Maligranda, L.: Some remarks on s-convex functions. Aequ. Math. 48(1), 100–111 (1994)
Bernstein, F., Doetsch, G.: Zur Theorie der Konvexen Funktionen. Math. Ann. 76, 514–526 (1915)
Kilicman, A., Saleh, W.: Notions of generalized s-convex functions on fractal sets. J. Inequal. Appl. 2015, 312 (2015)
Kilicman, A., Saleh, W.: Some generalized Hermite–Hadamard type integral inequalities for generalized s-convex functions on fractal sets. Adv. Differ. Equ. 2015, 301 (2015)
Chen, G., Srivastava, H.M., Wang, P., Wei, W.: Some further generalizations of Hölder’s inequality and related results on fractal space. Abstr. Appl. Anal. 2014, Article ID 832802 (2014)
Vivas, M., Hernandez, J., Merentes, N.: New Hermite–Hadamard and Jensen type inequalities for h-convex functions on fractal sets. Rev. Colomb. Mat. 50(2), 145–164 (2016)
Sarikaya, M.Z., Budak, H.: Generalized Ostrowski type inequalities for local fractional integrals. Proc. Am. Math. Soc. 145(4), 1527–1538 (2017)
Sun, W.: On generalization of some inequalities for generalized harmonically convex functions via local fractional integrals. Quaest. Math. 42(9), 1159–1183 (2019). https://doi.org/10.2989/16073606.2018.1509242
Budak, H., Sarikaya, M.Z., Yildirim, H.: New inequalities for local fractional integrals. Iran. J. Sci. Technol. Trans. Sci. 41(4), 1039–31046 (2017)
Kirmaci, U.S., Ozdemir, M.E.: On some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Appl. Math. Comput. 153, 361–368 (2004)
Erden, S., Sarikaya, M.Z., Celik, N.: Some generalized inequalities involving local fractional integrals and their applications for random variables and numerical integration. J. Appl. Math. Stat. Inform. 12(2), 49–65 (2016)
Pearce, C.E.M., Pecaric, J.: Inequalities for differentiable mappings with application to special means and quadrature formulae. Appl. Math. Lett. 13(2), 51–55 (2000)
Acknowledgements
The authors would like to express their sincere thanks to the editor and anonymous reviewers for their helpful comments and suggestions.
Availability of data and materials
Not applicable.
Funding
The work was supported by the Natural Science Foundation of China (Grant Nos. 11971142, 61673169, 11871202, 11701176, 11626101, 11601485).
Author information
Authors and Affiliations
Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Abdeljawad, T., Rashid, S., Hammouch, Z. et al. Some new local fractional inequalities associated with generalized \((s,m)\)-convex functions and applications. Adv Differ Equ 2020, 406 (2020). https://doi.org/10.1186/s13662-020-02865-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-020-02865-w