Generalized Fractional Hadamard and Fejér–Hadamard Inequalities for Generalized Harmonically Convex Functions

Abstract

In this paper, we define a new function, namely, harmonically α , h − m -convex function, which unifies various kinds of harmonically convex functions. Generalized versions of the Hadamard and the Fejér–Hadamard fractional integral inequalities for harmonically α , h − m -convex functions via generalized fractional integral operators are proved. From presented results, a series of fractional integral inequalities can be obtained for harmonically convex, harmonically h − m -convex, harmonically α , m -convex, and related functions and for already known fractional integral operators.

Fejér–Hadamard Type Inequalities for (α, h-m)-p-Convex Functions via Extended Generalized Fractional Integrals

“…(vi) For α = m = 1, p < 0 and h(t) = t in Theorem 3 (ii), Theorem 6 (ii) [29] is achieved. (vii) For p = −1 in Theorem 3 (ii), Theorem 6 [25] is achieved. (viii) For p = 1 in Theorem 3 (i), the result for (α, h-m)-convex function is achieved.…”

“…Farid, in [21], gave the Hadamard and the Fejér-Hadamard inequalities for convex functions by using the fractional integral operators containing Mittag-Leffler functions. For other known and new classes of functions, the Hadamard and the Fejér-Hadamard fractional inequalities can be found in [22][23][24][25][26][27][28][29][30] and references therein.…”

New Hermite–Hadamard Type Inequalities in Connection with Interval-Valued Generalized Harmonically (h1,h2)-Godunova–Levin Functions

“…Definition 6 (see [39]). The function G : S → R is said to be a harmonically (m,h)-convex function, if ∀c, d ∈ S, m ∈ (0, 1] and u ∈ [0, 1], we have…”

Refinements of some fractional integral inequalities for refined $(\alpha ,h-m)$-convex function

“…The Hermite-Hadamard inequality was generalized by Riemann-Liouville fractional integrals of convex functions in [29,30]. There exist many other versions of Hermite-Hadamard inequality in literature for different kinds of fractional integrals, see [2,8,13,14,17,23,[31][32][33][34] and the references therein. In the following, we give fractional versions of Hermite-Hadamard inequalities for convex functions via Riemann-Liouville fractional integrals.…”

“…Here, motivated and inspired by the ongoing research (see [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]), essentially the abovementioned works, we intend to demonstrate a few novel as well as detailed generalizations using the Riemann-Liouville operator applied over established well-known Hermite-Hadamard inequalities. More precisely, we considered the Riemann-Liouville fractional integrals with monotonically increasing function that plays a crucial role in our study.…”