Some integral inequalities for m-convex functions via generalized fractional integral operator containing generalized Mittag-Leffler function

G Abbas,G Farid,Lishan Liu

doi:10.1080/23311835.2016.1269589

G Abbas, G Farid + Show 1 more

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https://doi.org/10.1080/23311835.2016.1269589

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Abstract

In this paper, we are interested to prove some Hadamard and Fejer–Hadamard-type integral inequalities for m-convex functions via generalized fractional integral operator containing the generalized Mittag-Leffler function. In connection with we obtain some known results.

Highlights

Convex functions play an important role in the study of mathematical analysis
A close generalization of convex functions is m-convex function introduced by Toader (1984)
For m = 1 the above definition becomes the definition of convex functions

Summary

Introduction

Convex functions play an important role in the study of mathematical analysis. A close generalization of convex functions is m-convex function introduced by Toader (1984). Definition 1 A function f :[0, b] → R, b > 0 is said to be m-convex function if for all x, y ∈ [0, b] and t ∈ [0, 1] f (tx + m(1 − t)y) ≤ tf (x) + m(1 − t)f (y) holds for m ∈ [0, 1]. For m = 1 the above definition becomes the definition of convex functions. A convex function f :I → R is equivalently defined by the Hadamard inequality.

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