Some General Integral Operator Inequalities Associated with φ -Quasiconvex Functions

Young Chel Kwun,Ghulam Farid,Shin Min Kang,Praveen Agarwal,Moquddsa Zahra

doi:10.1155/2021/6696602

Abstract

This paper deals with generalized integral operator inequalities which are established by using φ -quasiconvex functions. Bounds of an integral operator are established which have connections with different kinds of known fractional integral operators. All the results are deducible for quasiconvex functions. Some fractional integral inequalities are deduced.

Highlights

Introduction and PreliminariesConvex functions play a vital role in the theory of mathematical analysis
Many generalizations have been given for the convex function, for example, α-convex, m-convex, h-convex, (α, m)-convex, (h, m)-convex, s-convex, (s, m)-convex, φ-convex, and quasiconvex functions
We will use φ-quasiconvex functions to study the bounds of unified integral operators, and the established results are directly related with fractional integral operators in particular cases

Summary

Introduction and Preliminaries

Convex functions play a vital role in the theory of mathematical analysis. Many generalizations have been given for the convex function, for example, α-convex, m-convex, h-convex, (α, m)-convex, (h, m)-convex, s-convex, (s, m)-convex, φ-convex, and quasiconvex functions (see [1,2,3,4,5,6,7,8,9,10]). We will use φ-quasiconvex functions to study the bounds of unified integral operators, and the established results are directly related with fractional integral operators in particular cases. All the fractional integral operators defined in [11,12,13,14,15] satisfy the results of this paper for φ-quasiconvex functions, and the results of [16,17,18,19] are reproduced in special cases. In [11], Farid defined a unified integral operator and proved the boundedness, linearity, and continuity of these integrals It is given in the following definition. Et al proved the bounds of unified integral operators for quasiconvex functions stated in eorems 1 to 4. ≤ Ecλ,,ρθ,,kκ,χ􏼐Φ g(x) − g x0􏼁􏼁λ; p􏼑Ψ g(x) − g x0􏼁􏼁 · max􏼐􏼌􏼌􏼌􏼌f′ x0􏼁􏼌􏼌􏼌􏼌, 􏼌􏼌􏼌􏼌f′(x)􏼌􏼌􏼌􏼌􏼑. We use the notation max􏼈f y0􏼁, f y0􏼁 + φ f x0􏼁, f y0􏼁􏼁􏼉 Mfφ x0, y0􏼁. (19)

Main Results

Applications

Concluding Remarks