New convex functions in linear spaces and Jensen’s discrete inequality
© Chen; licensee Springer. 2013
Received: 20 March 2013
Accepted: 30 August 2013
Published: 7 November 2013
In the present paper we introduce a new class of s-convex functions defined on a convex subset of a real linear space, establish some inequalities of Jensen’s type for this class of functions. Our results in special cases yield some of the recent results on classic convex functions.
The research on convexity and generalized convexity is one of the important subjects in mathematical programming, numerous generalizations of convex functions have been proved useful for developing suitable optimization problems (see [1–3]). s-convex functions defined on a space of real numbers was introduced by Orlicz in  and was used in the theory of Orlicz spaces. s-Orlicz convex sets and s-Orlicz convex mappings in linear spaces were introduced by Dragomir in . Some properties of inequalities of Jensen’s type for this class of mappings were discussed.
Definition 1.1 
Remark If , then, by the above definition, we recapture the concept of convex sets in linear spaces.
Definition 1.2 
Note that for we recapture the class of convex functions.
In this paper we introduce another class of s-convex functions defined on convex sets in a linear space. Some discrete inequalities of Jensen’s type are also obtained.
2 The relations among s-convex functions, s-Orlicz convex functions and convex functions in linear spaces
Let X be a linear space and be a fixed positive number, let be a convex subset. It is natural to consider the following class of functions.
for all and with .
Remark 2.2 If , then, by the above definition, we recapture the concept of convex functions in linear spaces.
There exist s-convex mappings in linear spaces which are not convex for some (see the following Example 1).
When , every non-negative convex function defined on a convex set in a linear space is also an s-convex function. When , every non-positive convex function defined on a convex set in a linear space is also an s-convex function.
hence f is an s-convex function on X; however, f is not a convex function on X with .
Remark 2.4 There exist s-Orlicz convex functions in linear spaces which are not s-convex functions for some .
we deduce that , i.e., K is s-Orlicz convex in .
Define , , we will show that f is an s-Orlicz convex function, f is not an s-convex function, for K is not a convex subset when . Since , , but for with .
3 Properties of s-convex functions in linear spaces
We consider the following properties of s-convex functions in linear spaces.
is an s-convex function on K.
is an s-convex function on K for all .
The proof is omitted.
Then when either one of (1) and (2) is satisfied, which shows that is a convex subset of K. □
f is an s-convex function on K.
- (2)For every and non-negative real number with , we have that(3.1)
Proof (2) ⇒ (1). This is obvious.
(1) ⇒ (2). We will prove by induction over , . For , the inequality is obvious by Definition 2.1. Suppose that the above inequality is valid for all . For natural number n, let and with .
If there is some , then delete the number and for the remaining number, inequality (3.1) is obvious by using the inductive hypothesis.
and the theorem is proved. □
The following corollaries are different formulations of the above inequalities of Jensen’s type.
where , , f is an s-convex function on the convex set , , , where denotes the finite subsets of the natural number set N.
- (1), , we have that(3.2)
- (2), , we have(3.3)
that is, the mapping is monotonic non-decreasing in the first variable on .
and inequality (3.2) is proved.
(2) Suppose that , with and .
and inequality (3.3) is proved. □
where , .
We get the first inequality (1) in Theorem 3.10.
We get the second inequality (2) in Theorem 3.10.
i.e., inequality (4) is proved. □
The author has greatly benefited from the referee’s report. So I wish to express our gratitude to the reviewer and the associate editor SS Dragomir for their valuable suggestions which improved the content and presentation of the paper. The work was supported by the project of Chongqing Municipal Education Commission (No. KJ090732) and special fund project of Chongqing Key laboratory of Electronic Commerce & Supply Chain System (CTBU).
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