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# Generalized Convex Functions and Some Inequalities on Fractal Sets

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## Abstract

In the paper, we introduce the generalized convex function on fractal sets and study the properties of the generalized convex function. Based on these properties, we establish the generalized Jensen's inequality and generalized Hermite-Hadamard's inequality on fractal sets. Furthermore, some applications are given .

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... Definition 9 (see [42]). A function ( ) is called -convex on if the following inequality holds true: ...
... Theorem 10 (see [42]). Assume that ( ) is an -local differentiable function on . ...
... Theorem 11 (see [42]). Assume that ( ) is a local fractional continuous function. ...
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... On the fractal space, Yang studies the monotonicity of function by the local fractional derivative [15]. And, in [16] the authors introduce the generalized convex function and study the analysis properties of the generalized convex function on fractal sets. The generalized convex function on fractal sets R α (0 < α < 1) can be stated as: f : [a, b] ⊂ R → R α is said to be generalized convex on [a, b], if inequality ...
... Note that, when s = 1, the generalized s-convex function in both sense is generalized convex function, see [16]. ...
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... Many researchers contemplated the properties of a function on the fractal space and built numerous sorts of fractional calculus by utilizing distinctive approaches, see [25,26]. Mo et al. [27] defined the generalized convex function on fractal sets R (0 < ≤ 1) of real numbers and established generalized Jensen's and Hermite-Hadamard's inequalities for a generalized convex function in the concept of local fractional calculus. In (2017) Sun [28] introduced the concept of harmonic convex function on fractal sets R (0 < ≤ 1) of real numbers and gave some Hermite-Hadamard inequalities for a generalized harmonic function ( ∈ GHK (I)). ...
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... In [11], Mo et al. proved the following generalized Hermite-Hadamard inequality for generalized convex function: ...
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