Generalized Convex Functions on Fractal Sets and Two Related Inequalities

In the paper, we introduce the generalized convex function on fractal sets (0 1) R   of real line numbers 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. Furthermore, some applications are given.

The convexity of functions plays a significant role in many fields, for example, in biological system, economy, optimization, and so on [1,2].And many important inequalities are established for the class of convex functions.For example, Jensen's inequality and Hermite-Hadamard's inequality are the best known results in the literature, which can be stated as follows.
Inspired by these investigations, we will introduce the generalized convex function on fractal sets and establish the generalized Jensen's inequality and generalized Hermite-Hadamard's inequality related to generalized convex function.We will focus our attention on the convexity since a function  is concave if and only if − is convex.So, every result for the convex function can be easily restated in terms of concave functions.
The paper is organized as follows.In Section 2, we state the operations with real line number on fractal sets and give the definitions of the local fractional derivatives and local fractional integral.In Section 3, we introduce the definition of the generalized convex function on fractal sets and study the properties of the generalized convex functions.In Section 4, we establish the generalized Jensen's inequality and generalized Hermite-Hadamard's inequality on fractal sets.In Section 5, some applications are given on fractal sets by means of the generalized Jensen's inequality.
: the -type set of the real line numbers is defined as the set   =   ∪   .

Generalized Convex Functions
From an analytical point of view, we have the following definition.
It follows immediately, from the given definitions, that a generalized strictly convex function is also generalized convex.But, the converse is not true.And if these two inequalities are reversed, then  is called a generalized concave function or generalized strictly concave function, respectively.
Note that the linear function () =     +   ,  ∈  is generalized convex and also generalized concave.
We will focus our attention on the convexity since a function  is concave if and only if − is convex.So, every result for the convex function can be easily restated in terms of concave functions.
In the following, we will study the properties of the generalized convex functions.Theorem 8. Let  :  →   .Then  is a generalized convex function if and only if the inequality holds, for any And by the generalized convexity of , we get From the above formula, it is easy to see that Reversely, for any two points  1 ,  3 ( 1 <  3 ) on  ⊆ , we take  2 =  1 + (1 − ) 3 for  ∈ (0, 1).Then  1 <  2 <  3 and  = ( 3 −  2 )/( 3 −  1 ).Using the above inverse process, we have So,  is a convex function on  ⊆ .In the same way, it can be shown that  is a generalized convex function on  ⊆  if and only if for any  1 ,  2 ,  3 ∈  with  1 <  2 <  3 .
(2 → 3) Take  1 ,  2 ∈ .Without loss of generality, we can assume that  1 <  2 .Since  () is increasing in the interval , then applying the generalized local fractional Taylor theorem, we have where  ∈ ( 1 ,  2 ).That is to say, and  2 −  3 = ( 2 −  1 ).Then from the third condition, we have At the above two formulas, multiply   and (1 − )  , respectively; then we obtain So  is a generalized convex function on .
Corollary 10.Let  ∈  2 (, ).Then  is a generalized convex function (or a generalized concave function) if and only if for any  ∈ (, ).

Some Inequalities
So, the mathematical induction gives the proof of Theorem 11.

Corollary 13 (generalized Cauchy-Schwarz's inequality). Let
Proof.Take () =  2 .It is easy to see that For another part, we first note that if  is a generalized convex function, then, for  ∈ [0, 1], it yields By adding these inequalities we have Then, integrating the resulting inequality with respect to  over [0, 1], we obtain 1 Γ (1 + ) ∫  Note that it will be reduced to the class Hermite-Hadamard inequality if  = 1.

Applications of Generalized Jensen's Inequality
Using the generalized Jensen's inequality, we can get some inequalities.