Reverse Hermite-Hadamard’s inequalities using ψ-fractional integral

Abstract: Our purpose in this paper is to use ψ−Riemann-Liouville fractional integral operator which is the fractional integral of any function with respect to another increasing function to establish some new fractional integral inequalities of Hermite-Hadamard, involving concave functions. Using the concave functions, we establish some new fractional integral inequalities related to the Hermite-Hadamard type inequalities via ψ−Riemann-Liouville fractional integral operator.


Introduction
T he classical calculus of derivatives and integrals which involves integer orders is extended with fractional orders that belong to the real numbers. In last few decades, the fractional calculus theory receives more attention due to its significant applications in several scopes such as physics, fluid dynamics, computer networking, image processing, biology, signal processing, control theory and other scopes. Because of the importance of fractional calculus, many researchers have shown their intense interest. One of the prevalent approaches among researchers is the use of fractional derivatives and integral operators. As a consequence, several distinct kinds of fractional integrals and derivatives operators have been realized, such as the Liouville, Riemann-Liouville, Katugampola, Weyl types, Hadamard and some other types which can be found in Kilbas et al., [1].
Hilfer [2] in (2000), through his contribution to improve the fractional calculus, established a new fractional derivative operator for any real order δ, which gives the Caputo derivative and the Riemann-Liouville fractional operator. The primary concept and properties and more information of ψ-Riemann-Liouville fractional derivative and integral can be found in [1]. In (2017), Almeida [3], introduced ψ-Caputo fractional derivative and investigated its significant properties. Recently, in 2018, Sousa and Oliveira [4], introduced a generalization of many existing fractional derivative operators called ψ-Hilfer derivative.
The mathematical inequalities play a very reliable role in classical integral and differential equations as well as in the past few years, many of useful mathematical inequalities have been originated by many authors, see [5][6][7][8]. One of the most significant integral inequalities is that discovered by Hermite [9] and Hadamard [10] for convex function f as follows If f is a concave function then both inequalities in (1) are held in a reversed direction. For some historical of Hermite-Hadamard inequalities [11] and the references therein. In the last few decades, these inequalities have been received a considerable attention by many authors and several articles have appeared in the literature, see [12][13][14]. In 2010, Dahmani [15], studied the Hermite-Hadamard type inequalities for concave functions by means of Riemann-Liouville fractional integral. Sarikaya et al., in 2013 [16], gave the Hermite-Hadamard type inequalities for convex function using Riemann-Liouville fractional integral. In 2014, Set et al., established Hermite-Hadamard type inequalities for s-convex functions in the second sense proved by Dragomir et al., [17] and m-convex functions via fractional integrals. In (2015), Noor et al., [18], derived some quantum estimates for Hermite-Hadamard inequalities for q-differentiable quasi convex functions and q-differentiable convex functions. Liu et al., [19] in 2016, introduced some inequalities of Hermite-Hadamard type for MT-convex functions using classical integrals and Riemann-Liouville fractional integrals. In 2017, Agarwal et al., [20], obtained some Hermite-Hadamard type inequalities for convex functions via (k, s)−Riemann-Liouville fractional integrals. Muhammad A. Khan [21] in 2018, proved new Hermite-Hadamard inequalities for convex function, s−convex and coordinate convex functions by using conformable fractional integrals. Recently in 2019, a lots of researchers studied Hermite-Hadamard inequalities for several kinds of convexity of the functions, for more details we refer the readers to see [22,23]. Very recently, in 2020, Chudziak and Ołdak introduced notion of a co-ordinated (F, G)-convex function defined on an interval in R 2 .
The main objective of this paper is to establish some new fractional integral Hermite-Hadamard inequalities for concave functions by using ψ−Riemann-Liouville fractional integral operator. Moreover, we introduce some new fractional integral inequalities related to the Hermite-Hadamard inequalities via ψ−Riemann-Liouville fractional integral operator. The paper is organized as follows: In Section 2, we collect some notations, definitions, results and preliminary facts which are used throughout this paper. In Section 3, we present the reverse Hermite-Hadamard's inequalities for concave functions. In Section 4, we give some other related results of Hermite-Hadamard type inequalities which involving ψ−Riemann-Liouville fractional integral operator.

Basic definitions and tools
This section is dedicated for some basic definitions and properties of fractional integrals used to obtain and discuss our new results. We also outline some basic results related to this work. Let , be a finite or infinite interval. Assume that f be an integrable function defined on Υ and ψ : Υ → R be an increasing function for all t ∈ Υ, which belong to C 1 (Υ, R) with condition that ψ (t) must be nonzero along the interval Υ. The ψ−Riemann-Liouville fractional derivative of order δ of a function f are defined by [1,24]: and Definition 1. Let δ > 0 and f be an integrable function defined on Υ and ψ (t) ∈ C 1 (Υ, R) be an increasing function such that ψ (t) = 0 for all t ∈ Υ. The left and right ψ−Riemann-Liouville fractional integral of order δ with respect to the function ψ of a function f are respectively defined by [1,24]: and Definition 2. The function f : (Λ ⊆ R) −→ R is said to be concave function if the following inequality holds for all x, y ∈ Λ and λ ∈ [0, 1] . We say that f is convex if the inequality (6) is reversed.

Theorem 4.
[15] Let f and g be two positive functions on [0, ∞). If f and g are a concave functions on [0, ∞), then for all p > 1, q > 1 and δ > 0, the following inequality holds:

The reverse Hermite-Hadamard's inequalities for fractional integral
Now, we give the reverse Hermite-Hadamard's inequalities involving concave functions for ψ-Riemann-Liouville fractional integral operators.
If ψ and f are concave and positive functions, then the following inequality holds: Proof. For any x, y ∈ [a, b] , using the concavity of f and ψ, we have Putting λ = 1 2 and using (9), we can write Let and where x, y are variables containing t. By substituting (11) and (12) in (10), we get Now, multiplying both sides of (13) by ψ (t) ψ δ−1 (t) , then integrating the resulting inequality with respect to t over [0, 1] , we obtain From (11), we have and So, we have Also from (12), we have and So, we have Using (14), (15) and (16), we get which is the first inequality in (8). To prove the second inequality and using the concavity of f and ψ, we can write for λ ∈ [0, 1] and Adding (18) and (19), we obtain Now, multiplying both sides of (20) by ψ (t) ψ δ−1 (t) , then integrating the resulting inequality with respect to t over [0, 1] , we obtain Hence, by combining the inequalities (17) and (20), we get the desired inequality (8).

Hermite-Hadamard type inequalities for fractional integral
In this section, we generalize some Hermite-Hadamard type inequalities involving concave functions introduced by Dahmani [15] using the Riemann-Liouville fractional integral with respect to other monotone and bijective function. In present part, we use only the left-sided fractional integrals (4). Moreover, we consider a = 0 to obtain and discuss our results. We first prove the following lemma: Proof. Since h be a concave function on Λ, so for any c, d ∈ [0, ∞) , we can write If we choose ψ (t) = λψ (c) + [1 − λ] ψ (d) , then we have Using the concavity of h, we obtain By (23) and (24), we get which is the required inequality (22). Then for all p > 1, q > 1 and δ > 0, the following inequality holds: Proof. Since f p and g q are a concave functions on Λ, so by Lemma (3), for any x, y > 0, we have and Multiplying both sides of (26) and (27) by ψ (y) Γ(δ) [ψ (x) − ψ (y)] δ−1 ψ δ−1 (y) , y ∈ (0, x) and integrating the resulting inequalities with respect to y over (0, x) , we obtain and Using the change of variable ψ (u) = ψ (x) − ψ (y) , where u ∈ [0, ∞) is a variable containing y, we have Then, we can write Now, by using (28) and (30), we get and using (29) and (31), we get The inequalities (32) and (33) yields On the other hand, we have f and g are positive functions and ψ is increasing function on [0, ∞). Then for any x > 0, p ≥ 1, q ≥ 1, we can write and Multiplying both sides of (35) and (36) by , then integrating the resulting inequalities with respect to y over (0, x) , we get and The inequalities (37) and (38) yields Combining the inequalities (34) and (39), we obtain the desired inequality (25).