Boundedness of Fractional Integral Operators Containing Mittag-Leffler Function via Exponentially s-Convex Functions

Abstract

The main objective of this paper is to obtain the fractional integral operator inequalities which provide bounds of the sum of these operators at an arbitrary point. These inequalities are derived for s-exponentially convex functions. Furthermore, a Hadamard inequality is obtained for fractional integrals by using exponentially symmetric functions. The results of this paper contain several such consequences for known fractional integrals and functions which are convex, exponentially convex, and s-convex.

On Strongly Convex Functions via Caputo–Fabrizio-Type Fractional Integral and Some Applications

“…ese estimates have a wider range of applications. Here, we shall propose some new estimates of the remainder term E(ϱ, d) which supplement, in a sense, those established in [30,[32][33][34][35].…”

“…It is clear that if the mapping f is not twice differentiable or the second derivative is not bounded on (a, b), then (49) cannot be applied. In recent studies [30,[32][33][34][35], Dragomir and Wang showed that the remainder term E(ϱ, d) can be estimated in terms of the first derivative only. ese estimates have a wider range of applications.…”

Hermite-Hadamard, Jensen, and Fractional Integral Inequalities for Generalized P -Convex Stochastic Processes

“…Our conclusions are applicable, since the expected value of a random variable is consistently bounded above by the expected value of the convex function of that random variable. It will be interesting to find parallel results by using the proposed definition in this study in the setting of other fractional integrals [14,15].…”

Inequalities for a Unified Integral Operator for Strongly α , m -Convex Function and Related Results in Fractional Calculus

“…Corollary 8. If we consider α = 1 and λ = 0 in (12), then the following inequality holds for m-convex functions: Lemma 10 (see [24]). Let f : ½a, b → ℝ be strongly ðα, mÞ -convex function, 0 < a < mb .…”

“…The objective of this paper is to obtain bounds of unified integral operators using strongly ðα, mÞ-convexity. It is defined as follows [24].…”