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New convex functions in linear spaces and Jensen’s discrete inequality
Journal of Inequalities and Applications volume 2013, Article number: 472 (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 
Let X be a linear space and . The set is called s-Orlicz convex in X if the following condition is true:
Remark If , then, by the above definition, we recapture the concept of convex sets in linear spaces.
Definition 1.2 
Let X be a linear space and . Let be an s-Orlicz convex set. The mapping is called s-Orlicz convex on K if for all and with , one has the inequality
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.
Definition 2.1 The mapping is called s-convex on K if
for all and with .
Remark 2.2 If , then, by the above definition, we recapture the concept of convex functions in linear spaces.
Remark 2.3 s-convex functions defined on a convex set in linear spaces are different from convex functions.
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.
Example 1 Let X be a normed linear space, let and , define for all . For every and with , when , either or , therefore, , and
when , from the monotonous increasing of the function in the interval , we have , , then
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 .
Example 2 , . Consider the set . Let , and let with . Then
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.
Theorem 3.1 Let be a convex set, be s-convex functions, . Then
is an s-convex function on K.
is an s-convex function on K for all .
The proof is omitted.
Proposition 3.2 Let be an s-convex function on K and so that is nonempty. Then is a convex subset of K when either one of the following two conditions is satisfied:
Proof Let and so that . Then and which imply that
Then when either one of (1) and (2) is satisfied, which shows that is a convex subset of K. □
Theorem 3.3 Let X be a linear space, let be a convex set, mapping , the following statements are equivalent:
f is an s-convex function on K.
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.
Now suppose , , let , , , , since . Using the inductive hypothesis, we have
and the theorem is proved. □
The following corollaries are different formulations of the above inequalities of Jensen’s type.
Corollary 3.4 Let be an s-convex function, be non-negative real numbers, . For every , we have that
Corollary 3.5 Let be an s-convex function, , then we have
Corollary 3.6 Let be an s-convex function, . For every non-negative real number , when , then we have
We consider the following function:
where , , f is an s-convex function on the convex set , , , where denotes the finite subsets of the natural number set N.
Theorem 3.7 Let be an s-convex function, , , , then
, , we have that(3.2)
, , we have(3.3)
that is, the mapping is monotonic non-decreasing in the first variable on .
Proof (1) Let , . Then we have
and inequality (3.2) is proved.
(2) Suppose that , with and .
With the above assumptions, consider the sequence
whence we get
and inequality (3.3) is proved. □
With the above assumptions, consider the sequence
Corollary 3.8 With the above assumptions,
i.e., the sequence is non-decreasing and one has the inequality
Theorem 3.9 Let be an s-convex function, and so that . Let , . Then one has the inequalities
where , .
Theorem 3.10 Let be an s-convex function, and so that . Then, for every , with , we have
Proof By the s-convexity of f, we can state
We get the first inequality (1) in Theorem 3.10.
By the following inequality
We get the second inequality (2) in Theorem 3.10.
Let , by the inequality in Theorem 3.9, we have
i.e., inequality (3) is obtained.
i.e., inequality (4) is proved. □
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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).
The author declares that they have no competing interests.
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Chen, X. New convex functions in linear spaces and Jensen’s discrete inequality. J Inequal Appl 2013, 472 (2013). https://doi.org/10.1186/1029-242X-2013-472
- s-Orlicz convex functions
- s-convex functions
- Jensen’s inequalities