Trapezium-Type Inequalities for an Extension of Riemann–Liouville Fractional Integrals Using Raina’s Special Function and Generalized Coordinate Convex Functions

Abstract

In this paper, the authors analyse and study some recent publications about integral inequalities related to generalized convex functions of several variables and the use of extended fractional integrals. In particular, they establish a new Hermite–Hadamard inequality for generalized coordinate ϕ-convex functions via an extension of the Riemann–Liouville fractional integral. Furthermore, an interesting identity for functions with two variables is obtained, and with the use of it, some new extensions of trapezium-type inequalities using Raina’s special function via generalized coordinate ϕ-convex functions are developed. Various special cases have been studied. At the end, a brief conclusion is given as well.

Some Hermite–Hadamard-Type Fractional Integral Inequalities Involving Twice-Differentiable Mappings

“…In the field of fractional calculus, several mathematicians worked on the concept of h-convexity and presented different types of Hermite-Hadamard type inequalities. For the readers, see [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36] and the references cited therein.…”

“…For some recent generalization of Hermite-Hadamard type inequalities via fractional operators, readers can refer to [26][27][28][29][30] and the references cited therein. Recently, in [27,28], the authors introduced a new class of convex functions and presented the Hermite-Hadamard inequality using a generalized Riemann-Liouville fractional integral operator concerning a monotonic function.…”

“…Recently, in [27,28], the authors introduced a new class of convex functions and presented the Hermite-Hadamard inequality using a generalized Riemann-Liouville fractional integral operator concerning a monotonic function. Cortez et al in [29] presented some trapezium-type inequalities for generalized coordinated convex function via a new form of the Riemann-Liouville fractional operator using Raina's special function. Kashuri et al [30,31] worked on the k-Riemann-Liouville fractional operator to study the inequalities of Hermite-Hadamard type.…”

Hermite–Jensen–Mercer-Type Inequalities via Caputo–Fabrizio Fractional Integral for h-Convex Function

“…For convex functions, the Hermite-Hadamard inequality is a famous inequality that has been proved in many ways and has several extensions and generalizations in the literature (see [9][10][11][12][13][14][15][16][17][18][19]). The Hermite-Hadamard inequality for the convex function is defined as:…”

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“…Similarly, the concept of convexity has had a great evolution because of its wide application in various fields of science, including those where the fractional and quantum calculus are applied [5][6][7][8][9][10][11]. In the last decades generalizations of the convexity, such as log-convexity, s−convexity in the first and second sense , Wright convexity, E−convexity, m−convexity, ϕ−convexity, GA−convexity, (s, ϕ)−convexity and others [12][13][14][15][16][17][18][19][20][21][22][23] have arisen.…”