Generalized fractional inequalities for quasi-convex functions

Abstract

The class of quasi-convex functions contain all those finite convex functions which are defined on finite closed intervals of real line. The aim of this paper is to establish the bounds of the sum of left and right fractional integral operators using quasi-convex functions. An identity is formulated which is used to find Hadamard-type inequalities for quasi-convex functions. Connections with some known results are analyzed. Furthermore, some implications are derived by considering some examples of quasi-convex functions.

On boundedness of unified integral operators for quasiconvex functions

“…All these results hold for almost all kinds of associated fractional and conformable integral operators. Also, some very particular cases of the proved results are already published in [4,9,27], and connection with them is stated in remarks.…”

“…Similarly, by putting x = a in (2.9) we obtain Remark 3 Theorem 2 provides the boundedness of all known operators defined in [2,3,6,10,13,14,18,20,21,23,25,26]. Especially, the boundedness of the integral operator given in Definition 4, which is studied in [27].…”

Study of fractional integral inequalities involving Mittag-Leffler functions via convexity

“…The extended Mittag-Leffler function (1.3) produces the related functions defined in [23,24,[26][27][28], see [29,Remark 1.3].…”

Estimations of Riemann–Liouville k-fractional integrals via convex functions

“…In fact these formulations of fractional integral operators have been established by Letnikov [10], Sonin [12], and then by Laurent [9]. A lot of fractional integral inequalities have been established in literature (for more details, see [1,3,4,5,6,7,8,13]).…”