Some new Hermite-Hadamard type inequalities for generalized harmonically convex functions involving local fractional integrals

Abstract: In this paper, we establish a new integral identity involving local fractional integral on Yang’s fractal sets. Using this integral identity, some new generalized Hermite-Hadamard type inequalities whose function is monotonically increasing and generalized harmonically convex are obtained. Finally, we construct some generalized special means to explain the applications of these inequalities.


Introduction
Let φ : I ⊆ R → R be a convex function and α, β ∈ I with α < β, then the following inequality holds, which is well known as Hermite-Hadamard's inequality [1] for convex functions. Both inequalities hold in the reversed direction if φ is concave. Convex function is an important function in mathematical analysis and has been applied in many aspects [2,3]. With the extension of the definition of convex function, Hermite-Hadamard's inequality has been deeply studied.
In [11],İşcan gave the definition of harmonically convexity as follows: Definition 1. Let I ⊂ R \ {0} be a real interval. A function φ : I → R is said to be harmonically convex, if for all x, y ∈ I and t ∈ [0, 1]. If the inequality in (1.2) is reversed, then φ is said to be harmonically concave.
In [22], Sun introduced the definition of the generalized harmonically convex function on Yang's fractal sets as follows: Definition 2. Let I ⊂ R \ {0} be a real interval. A function φ : I → R (0 < ≤ 1) is said to be generalized harmonically convex, if for all x, y ∈ I and t ∈ [0, 1]. If the inequality in (1.3) is reversed, then φ is said to be generalized harmonically concave. The sign represents the fractal dimension.
Based on the theory of local fractional calculus and the definition of the generalized harmonically convex function on Yang's fractal sets, the main aim of this paper is using a new integral identity and monotonicity of functions to establish some new Hermite-Hadamard type inequalities involving local fractional calculus.

Preliminaries
Let R (0 < ≤ 1) be -type set of the real line numbers on Yang's fractal sets, and give the following operation rules, see [17,18]. The sign represents the fractal dimension, not the exponential sign.

Main results
For convenience, we use the symbol A t to denote tα + (1 − t)β in the following sections. Lemma 6. Let I ⊂ (0, ∞) be an interval, φ : I • → R (I • is the interior of I) such that φ ∈ D (I • ) and φ ( ) ∈ C (α, β) for α, β ∈ I • with α < β. Then the following equality holds where Proof. Calculating I 1 , I 2 , from Lemma 1(1), we get and Calculating I 3 , by the local fractional integration by parts, we have Using changing variable with x = αβ A t , we have Adding I 1 − I 3 , the desired result is obtained. This completes the proof.
Proof. Since φ is an increasing function on I • , and 0 < α < 2αβ α+β < β, we can obtain From the proof of Lemma 6, we have Taking modulus in equality (3.1), we obtain From Lemma 6, using the property of the modulus and the generalized Hölder's inequality, we have Since |φ ( ) | q is generalized harmonically convex on [α, β], thus By calculating, we get Similarly, we get and From (3.4)-(3.9), we get inequality (3.2). This completes the proof.
, the following local fractional integrals inequality holds.

Applications to special means
We consider the following -type generalized special means of the real line numbers α , β with α < β on Yang's fractal sets.

Conclusions
In this paper, the research on Hermite-Hadamard type inequalities is extended to Yang's fractal space. By using the definitions of generalized harmonically convex function and the theory of local fractional calculus, we construct some new Hermite-Hadamard type integral inequalities for monotonically increasing functions with generalized harmonically convexity. Some applications related to the special mean are established by using the obtained inequalities, which shows that our results have certain application significance. Our research may inspire more scholars to further explore Hermite-Hadamard type integral inequalities on Yang's fractal sets.