Abstract
In this paper we have derived the fractional integral inequalities by defining exponentially (s,m)-convex functions. These inequalities provide upper bounds, boundedness, continuity, and Hadamard type inequality for fractional integrals containing an extended Mittag-Leffler function. The results about fractional integral operators for s-convex, m-convex, (s,m)-convex, exponentially convex, exponentially s-convex, and convex functions are direct consequences of presented results.
Highlights
1 Introduction Convex functions are very useful in mathematical analysis due to their fascinating properties and convenient characterizations
3 Concluding remarks This paper has investigated generalized fractional integral inequalities which provide the bounds of fractional integral operators containing Mittag-Leffler functions in their kernels
By setting different values of parameters involved in the Mittag-Leffler function, the results for various known fractional operators can be obtained
Summary
Convex functions are very useful in mathematical analysis due to their fascinating properties and convenient characterizations.Definition 1 A function f : I → R is said to be convex function if the following inequality holds:f ta + (1 – t)b ≤ tf (a) + (1 – t)f (b) (1.1)for all a, b ∈ I and t ∈ [0, 1]. Convex functions are very useful in mathematical analysis due to their fascinating properties and convenient characterizations. Definition 1 A function f : I → R is said to be convex function if the following inequality holds:. If inequality (1.1) holds in reverse order, the function f is called concave function. A graphical interpretation of a convex function f over an interval [a, b] provides at a glance the following well-known Hadamard inequality:. This inequality has been studied extensively, and a lot of its versions have been published by defining new functions obtained from inequality (1.1). Qiang et al Journal of Inequalities and Applications
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