New Hermite–Hadamard Type Inequalities for ψ-Riemann–Liouville Fractional Integral via Convex Functions

Hermite–Hadamard integral inequality provides the estimate for the integral average of any convex function defined on a compact interval. In recent years, manymathematicians extended it to s-convex functions, quasi-convex functions, ψ-Riemann–Liouville convex functions, η-convex functions, r-convex functions, (α, m)-convex functions, and Lipschitzian functions, see [1–7]. -e most famous are trapezoid inequality and midpoint inequality [8, 9]: suppose that f: [a, b]⟶ R is a differentiable function on (a, b) with a< b. If |f′| is convex on [a, b], then


Introduction
Hermite-Hadamard integral inequality provides the estimate for the integral average of any convex function defined on a compact interval. In recent years, many mathematicians extended it to s-convex functions, quasi-convex functions, ψ-Riemann-Liouville convex functions, η-convex functions, r-convex functions, (α, m)-convex functions, and Lipschitzian functions, see [1][2][3][4][5][6][7]. e most famous are trapezoid inequality and midpoint inequality [8,9]: suppose that f: (1) In [10], Mehrez e classical integro-differential equation has been studied deeply, and a rigorous and systematic theoretical system has been established. In recent decades, the theoretical research of fractional integro-differential equation has been constantly improved. Many theoretical achievements have been obtained [2-4, 7, 11-15]. e research of Hermite-Hadamard type inequalities for fractional integrals has also been deepened.
In [3], Liu et al. established the Hermite-Hadamard type inequalities for ψ-Riemann-Liouville fractional integrals via convex functions: let g: [a, b] ⟶ R be a positive convex integrable function, 0 ≤ a < b. ψ(x) is an increasing positive monotone function and have continuous derivation on (a, b], and α ∈ (0, 1). en, the following inequality holds: (3) is an increasing positive monotone function on (a, b] and have continuous derivation on (a, b), and α ∈ (0, 1). en, the following inequalities hold: where I α:ψ ψ − 1 (a) + and I α:ψ ψ − 1 (b) − are the left-sided and right-sided ψ-Riemann-Liouville fractional integral operators of order α, which are listed in Section 2. e aim of this paper is to extend the range of argument of the function g ′ and establish some new Hermite-Hadamard type inequalities for ψ-Riemann-Liouville fractional integrals via convex functions on the basis of (3)- (5). en, we apply them to special means of real numbers and construct inequalities for the beta function.

Preliminaries
In this section, we recall the definition of ψ-Riemann-Liouville fractional integral and some lemmas about (Γ(α + 1) Definition 1 (see [16]). Let (a, b)(− ∞ ≤ a < b ≤ ∞) be a finite or infinite interval of the real line R and α > 0. Also, let ψ(x) be an increasing and positive monotone function on (a, b], having a continuous derivative ψ ′ (x) on (a, b). e left-sided and right-sided ψ-Riemann-Liouville fractional integrals of order α for function f with respect to another function ψ on [a, b] are defined by respectively. Here, Γ(α) is the gamma function.

Main Results
First, we give an estimate for (Γ(α + 1) is an increasing and positive monotone function on [a, b], with a continuous derivative ψ ′ (x) on (a, b), 0 ≤ a < b, and α ∈ (0, 1). If g is a convex function on [a, b], then the following inequalities hold: Proof. From the left side of (3), we can directly get the left side of (11). en, we prove the right side of (11).
Discrete Dynamics in Nature and Society 3 Let we have From (12)- (15), we have the inequality From (3), we have 4 Discrete Dynamics in Nature and Society From (16) and (17), we obtain the right-hand side of inequality (11).
is completes the proof.
□ Discrete Dynamics in Nature and Society