General Raina fractional integral inequalities on coordinates of convex functions

Dumitru Baleanu,Pshtiwan Othman Mohammed,Artion Kashuri,Badreddine Meftah

doi:10.1186/s13662-021-03241-y

Dumitru Baleanu, Pshtiwan Othman Mohammed + Show 2 more

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https://doi.org/10.1186/s13662-021-03241-y

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Abstract

Integral inequality is an interesting mathematical model due to its wide and significant applications in mathematical analysis and fractional calculus. In this study, authors have established some generalized Raina fractional integral inequalities using an (l_{1},h_{1})-(l_{2},h_{2})-convex function on coordinates. Also, we obtain an integral identity for partial differentiable functions. As an effect of this result, two interesting integral inequalities for the (l_{1},h_{1})-(l_{2},h_{2})-convex function on coordinates are given. Finally, we can say that our findings recapture some recent results as special cases.

Highlights

In the past two decades, fractional calculus has received much attention
Most of the studies have been devoted to the existence and uniqueness of solutions for fractional differential equations (FDEs); see e.g. [4,5,6,7,8,9]
It is important and necessary to specify which model or definition is being used because there are many different ways of defining fractional integrals and derivatives

Summary

Introduction

In the past two decades, fractional calculus has received much attention. The fast interest in the topic is due to its extensive applications in various fields such as biochemistry, physics, viscoelasticity, fluid mechanics, computer modeling, and engineering, see [1,2,3] for further detail. Definition 1.1 ([1, 2]) For any L1 function f (x) on an interval [χ1, χ2] with x ∈ [χ1, χ2], the ηth left-RL fractional integral of f (x) is defined as follows: RLJηχ1+f (x) :=. Definition 1.2 ([30]) For any L1 function f (x) on an interval [χ1, χ2] with x ∈ [χ1, χ2], the ηth left Raina fractional integral of f (x) is defined as follows:. × Fσρ22,η2 ω2(y – ξ2)ρ1 φ(ξ1, ξ2) dξ dξ1, and χ2 χ4 σ ρ ,η,χ2– ,χ4– ,ω φ (ξ1 – x)η1–1(ξ2 – y)η2–1Fσρ11,η1 ω1(ξ1 – x)ρ1 y. Motivated by the above results, in this paper we establish some generalized integral inequalities using an (l1, h1)-(l2, h2)-convex function on coordinates. As an effect of this result, two interesting integral inequalities for an (l1, h1)-(l2, h2)-convex function on coordinates are given. A brief conclusion is provided as well

Main results

Conclusion