On Ostrowski–Mercer’s Type Fractional Inequalities for Convex Functions and Applications

Abstract

This research focuses on the Ostrowski–Mercer inequalities, which are presented as variants of Jensen’s inequality for differentiable convex functions. The main findings were effectively composed of convex functions and their properties. The results were directed by Riemann–Liouville fractional integral operators. Furthermore, using special means, q-digamma functions and modified Bessel functions, some applications of the acquired results were obtained.

On Fractional Ostrowski-Mercer-Type Inequalities and Applications

“…When ω = 1 in ( 5) and ( 6), then they coincides with (3) and ( 4), respectively. They also coincide with the Hadamard fractional integral [43] by setting µ = 0 and ω → 0 in (5) and κ = 0 and ω → 0 in (6). In addition, by choosing µ = 0 in (5) and κ = 0 in (6), we have the generalized fractional integrals [44].…”

“…By utilizing this inequality, He Chengtian calculated the approximate values of the fractional day of a moon and a year. Over the course of time, researchers have broadened the scope of convex mappings, leading to the discovery of various variants of the Hermite-Hadamard inequality, see [3][4][5][6][7][8][9][10][11][12][13][14].…”

Hermite–Hadamard–Mercer-Type Inequalities for Three-Times Differentiable Functions

“…Starting from the results of [19], in Section 4, and motivated by recent studies such as [16,17], several Hermite-Hadamard-Mercer-type inequalities are presented in the frame of fractional integrals, for functions whose third derivative, in absolute values, is convex in Theorems 9-11. For that a key result is given in Lemma 2.…”

“…This inequality has been studied by many scholars, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], and references therein. From the study of the Hermite-Hadamard inequality the Ostrowski, Simpson, midpoint, and trapezoidal inequalities were also obtained, as well as many Hermite-Hadamard-like inequalities [16,17]. Some studies on Hermite-Hadamard-like inequalities for functions whose third derivatives in absolute value are convex can also be found in [18][19][20].…”

Several Quantum Hermite–Hadamard-Type Integral Inequalities for Convex Functions

“…q-calculus concepts [26] on finite intervals was used to find quantum analogues of known mathematical definitions and results. New quantum analogues [27] of the Ostrowski inequalities [28], using first-order quantum differentiable convex mappings, were presented by Noor et al Several bounds for the left-hand side (LHS) of quantum H-H inequalities [29] were established. New quantum analogues of the classical Simpson's inequality were presented [30] for pre-invex functions.…”