The Hermite–Hadamard–Jensen–Mercer Type Inequalities for Riemann–Liouville Fractional Integral

Abstract

In this paper, we give Hermite–Hadamard type inequalities of the Jensen–Mercer type for Riemann–Liouville fractional integrals. We prove integral identities, and with the help of these identities and some other eminent inequalities, such as Jensen, Hölder, and power mean inequalities, we obtain bounds for the difference of the newly obtained inequalities.

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

“…The Hermite-Hadamard-Mercer inequalities have been improved and generalized in many directions in recent years by many researchers; see, for example, [26][27][28][29][30][31][32]. Thus, in [26,27] new variants of Hermite-Hadamard-Mercer inequalities are established for Riemann-Liouville fractional integrals, and in [28,29] Hermite-Hadamard-Mercer-type inequalities for generalized fractional integrals and for fractional integrals and differentiable functions are proven.…”

“…The Hermite-Hadamard-Mercer inequalities have been improved and generalized in many directions in recent years by many researchers; see, for example, [26][27][28][29][30][31][32]. Thus, in [26,27] new variants of Hermite-Hadamard-Mercer inequalities are established for Riemann-Liouville fractional integrals, and in [28,29] Hermite-Hadamard-Mercer-type inequalities for generalized fractional integrals and for fractional integrals and differentiable functions are proven. New estimate bounds are given for Ostrowski-type inequalities by using the Jensen-Mercer inequality in [30] and the concept of convexity for intervalvalued functions was used to find fractional Hermite-Hadamard-Mercer-type inequalities in [31].…”

Hermite–Hadamard–Mercer Inequalities Associated with Twice-Differentiable Functions with Applications

“…After that, many researchers tended towards these useful inequalities and succeeded in proving different new variants of Hermite-Hadamard-Mercer inequalities. For example, in [9][10][11], the authors applied the Riemann-Liouville fractional integrals and established Hermite-Hadamard-Mercer-type inequalities with their estimates for differentiable convex functions. In [12], Set et al demonstrated some new Hermite-Hadamard-Mercer-type inequalities for generalized fractional integrals, and each inequality demonstrated here is a family of inequalities for different fractional operators.…”

Some new q-Hermite-Hadamard-Mercer inequalities and related estimates in quantum calculus

“…For more recent inequalities related to (1.2) and (1.3), one can consult [5,6,7,8,9,10]. On the other hand, quantum calculus is an important branch of calculus and it has a wide range of applications in the fields of mathematics and physics.…”