Hermite–Hadamard-type inequalities for η h -convex functions via ψ -Riemann–Liouville fractional integrals

In this paper, we establish some new Hermite–Hadamard type inequalities involving ψ -Riemann–Liouville fractional integrals via η h -convex functions. Finally, we give some applications to the special means of real numbers.


Introduction
The study of fractional calculus, differential equation and inequalities got rapid development in the last few decades. Comparatively, fractional derivatives and integrals express physical phenomena and hereditary properties of various materials in a more precise way than classical derivatives. Classical derivatives are not enough to solve modern problems of engineering, physics, and other applied sciences because of involvement of fractional equations and inequalities.
To overcome this difficulty, many researchers are working on this area of research, see e.g. [1][2][3][4]. For more about the topic, we refer to the book [5].
The Hermite-Hadamard type inequalities are considered as one of important inequalities in convex analysis.
The function g : ϕ → R is convex if the following inequality holds: for all β, γ ∈ ϕ and t ∈ (0, 1). Hermite-Hadamard type inequalities have been studied extensively by many researchers, and a significant number of generalizations have appeared in a number of papers on convex analysis, inequality theory, and fractional integrals (see e.g. [6][7][8][9]).
The present paper is organized as follows: In the second section we provide some preliminary material and basic lemmas. The third section is devoted to the main results, whereas in the last section we give some applications to means.

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for all β, γ ∈ ϕ, t ∈ [0, 1] and h : J → R is a nonnegative function.
be a finite or infinite interval on the real line R, and let α > 0. Also, let ψ(x) be an increasing positive function on (β, γ ] with continuous derivative ψ (x) on (β, γ ). Then the left-and right-sided ψ-Riemann-Liouville fractional integrals of a function g with respect to the function ψ on [β, γ ] are defined by The next remark provides the relations among convexities.

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(2.12)

Main results
We are now in a position to establish some inequalities of Hermite-Hadamard type involving ψ-Riemann-Liouville fractional integrals with α ∈ (0, 1) via η h convex functions.
Then we have the following inequality for fractional integrals: where M is an upper bound of η, Multiplying both sides of (3.3) by t α-1 and then integrating the resulting inequality with respect to t over [0, 1], we obtain

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where (3.4) is used, so the left-hand side inequality in (3.1) is proved.
To prove the right-hand side inequality in (3.1), since g is a η h convex function, then for t[0, 1] we have

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Multiplying both sides of (3.5) by t α-1 and then integrating, we obtain So then The proof is completed.

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The proof is completed.

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Proof Using Lemma (2.13) and the Holder inequality via the η h -convexity of |g | q (q > 1),

Park et al. Advances in Difference Equations
(2020) 2020:602 Page 9 of 14 The proof is completed.

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Proof Using Lemma (2.13) and the power-mean inequality via the η h -convexity of |g | q (q > 1), we have

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This completes the proof.
Remark 3.9 If we take h(t) = t and η(β, γ ) = βγ , then Theorem 3.8 will be reduced as a result of classical convexity.
Now we give some applications to the special means of a real number.

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Proof Applying Theorem 3.3, Corollary 3.7, and Corollary 3.10 respectively for g(x) = 1 x , ψ)(x) = x, and α = 1, we immediately obtain the result.

Conclusion
In this article we established the Hermite-Hadamard type inequalities for η h convex functions. The main motivation of the article is [16]. The Hermite-Hadamard inequality derived here involved ψ-Riemann-Liouville fractional integrals. We also give some application of our results. Hopefully, the idea used in this paper will be interesting for the research of integral inequality and fractional calculus.