Hermite-Hadamard Type Inequalities for Generalized Convex Functions on Fractal Sets Style

We aim to  establish certain generalized Hermite-Hadamard's inequalities for generalized convex functions via local fractional integral. As special cases of some of the results presented here, certain interesting inequalities involving generalized arithmetic and logarithmic means are obtained.


Introduction and Preliminaries
In order to describe the definition of the local fractional derivative and local fractional integral, recently, one has introduced to define the following sets (see, e.g., [15,19]; see also [2]): For 0 < α ≤ 1, (i) the α-type set of integers Z α is defined by (ii) the α-type set of rational numbers Q α is defined by (iii) the α-type set of irrational numbers J α is defined by (iv) the α-type set of real line numbers R α is defined by R α := Q α ∪ J α .

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M. Tomar, P. Agarwal and J. Choi Here and in the following, let R, R + , Q, Z and N be the sets of real and positive real numbers, rational numbers, integers and positive integers, respectively, and and, whenever the α-type set R α of real line numbers is involved, the α is assumed to be tacitly 0 < α ≤ 1.
Proposition 1.1.Each of the following statements holds true: (i) Like the usual real number system (R, +, •), (R α , +, •) is a field; (ii) The additive identity 0 α and the multiplicative identity 1 α are unique, respectively; (iii) The additive inverse element and the multiplicative inverse element are unique, respectively; (iv) For each a α ∈ R α , its inverse element (−a) α may be written as −a α ; for each b α ∈ R α \ {0 α }, its inverse element (1/b) α may be written as In order to introduce the local fractional calculus on R α , we begin with the concept of the local fractional continuity as in Definition 1.1.
Among several attempts to have defined local fractional derivative and local fractional integral (see [14, Section 2.1]), we choose to recall the following definitions of local fractional calculus (see, e.g., [3,14,15]): Definition 1.2.The local fractional derivative of f (x) of order α at x = x 0 is defined by ) and Γ is the familiar Gamma function (see, e.g., [13, Section 1.1]). Let provided the limit exists (in fact, this limit exists if Here, it follows that a I (α) x g exists for any x ∈ [a, b] and a function g : [a, b] → R α , then we denote We give some of the features related to the local fractional calculus that will be required for our main results (see [15]).
Lemma 1.2.The following identities hold true: For further details on local fractional calculus, one may refer to [14]- [18].
which is known as the Hermite-Hadamard inequality.
Mo et al. [8] introduced the following generalized convex function.
holds, then f is called a generalized convex function on I.
Here are two basic examples of generalized convex functions: (1) Recently the fractal theory has received a significant attention (see, e.g., [1,3,6,7,9,10,11,12]).Mo et al. [8] proved the following analogue of the Hermite-Hadamard inequality (1.4) for generalized convex functions: Remark 1.3.The double inequality (1.5) is known in the literature as generalized Hermite-Hadamard integral inequality for generalized convex functions.Some of the classical inequalities for means can be derived from (1.5) with appropriate selections of the mapping f .Both inequalities in (1.4) and (1.5) hold in the reverse direction if f is concave and generalized concave, respectively.For some more results which generalize, improve and extend the inequalities (1.5), one may refer to the recent papers [1,6,7], [9]- [11] and references therein.
An analogue in the fractal set R α of the classical Hölder's inequality has been established by Yang [15], which is asserted by the following lemma.
Here, in this paper, we establish certain generalized Hermite-Hadamard's inequalities for generalized convex functions via local fractional integral.As special cases of some of the results presented here, certain interesting inequalities involving generalized arithmetic and logarithmic means are obtained.

Main Results
Lemma 2.1.Let I ⊆ R be an interval, f : the following identity holds true: Proof: Using the local fractional integration by parts, we have Similarly, we also get the following identities: Hermite-Hadamard Type Inequalities 107 and Proof: Using Lemma 2.1 and taking the modulus, we have (2.7) If we use generalized convexity of f (α) on [a, b], we get from the inequality (2.7) that (2.11) Theorem 2.4.Suppose that assumptions of Lemma 2.1 are satisfied.If f (α) q is generalized convex on [a, b] for some fixed q > 1, then the following inequality holds Hermite-Hadamard Type Inequalities 109 true: (2.12) where Proof: From Lemma 2.1 and using generalized Hölder integral inequality, we have (2.15) Similarly,

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If we substitute the inequalities from (2.15) to (2.18) in the inequality (2.14), we obtain the desired inequality (2.12).Note that we have also used the following identity: .
This completes the proof.✷ Taking x = a+b 2 in the result in Theorem 2.4 and using the convexity of f (α) q , we get an inequality asserted in Corollary 2.5.
Corollary 2.5.The following inequality holds true: where A and B are given as in (2.13).
Theorem 2.7.Suppose that assumptions of Lemma 2.1 are satisfied.If f (α) q is generalized convex on [a, b] for some fixed q ≥ 1, then the following inequality holds true:

.22)
Proof: Suppose that q ≥ 1.From Lemma 1.1 and using the well-known powermean inequality, we have where k and L are given as in (2.22).

Applications
Here we apply some of the results in the previous section to the following generalized means (see [3,12]):

Bibliography
Further, for this partition P , let ∆t := max 0≤j≤N −1 ∆t j where ∆t j := t j+1 −t j {j = 0, . . ., N − 1}.Then the local fractional integral of f on the interval [a, b] of order α (denoted by a