# General Convexity of Some Functionals in Seminormed Spaces and Seminormed Algebras

- Todor Stoyanov
^{1}Email author

**2010**:643768

https://doi.org/10.1155/2010/643768

© Todor Stoyanov. 2010

**Received: **30 July 2010

**Accepted: **27 October 2010

**Published: **28 October 2010

## Abstract

We prove some results for convex combination of nonnegative functionals, and some corollaries are established.

## Keywords

## 1. Introduction

Inequalities have been used in almost all the branches of mathematics. It is an important tool in the study of convex functions in seminormed space and seminormed algebras. Recently some works have been done by Altin et al. [1, 2], Tripathy et al. [1–6], Tripathy and Sarma [3, 4], Chandra and Tripathy [5], Tripathy and Mahanta [6], and many others involving inequalities in seminormed spaces and convex functions like the Orlicz function.

In this paper, inequalities for convex combinations of functionals satisfying conditions (a) and (b) are formulated in the theorems, and some corollaries are proved, using the theorems. Condition (a) relates to nonnegative functionals over which the inequalities in Theorems 1.1 and 1.4 on seminorm are proved. In Theorem 1.1, we consider seminormed spaces, and in Theorem 1.4 seminormed algebras. Condition (b) relates generally to the representations between seminormed spaces and seminormed algebras. The inequalities formulated in this way are proved in Corollaries 1.2 and 1.5. In this paper we consider the following generalization of the convexity in seminormed algebras. , where , for , is the norm in , and is a real number.

In order to justify our study, we have provided an example related to real functions of one variable, similar examples can be constructed. This has been used in the geometry of Banach spaces as found in [7, 8]. Similar statements related to functionals in finite-dimensional spaces and countable dimensional spaces have been provided in [9]. These results can be applied in the mentioned areas.

Theorem 1.1.

Let be a seminormed space over and the nonnegative functional f satisfy the following condition:

(a) , for all x,y with , where are nondecreasing functions such that . Then,

(2)the functions and are convex. Then, if , , , for , the inequality is satisfied.

Proof.

There exists in compliance with (1). Therefore .

Let us consider elements , , and we suppose .

Using the principle of induction over , we will probe that .

- (1)

Applying the induction, we get .

Corollary 1.2.

Let and be seminormed spaces over and . Then in Theorem 1.1, one replaces condition (a) by condition (b): , for all with , and all the rest of the conditions are satisfied. Then, with , , , , the inequality is satisfied.

Proof.

We consider the functional . Then, knowing (b), we conclude that satisfies Theorem 1.1's conditions and hence the needed inequality.

Example 1.3.

*,*, , , we will obtain the inequality

Proof.

when , that is, ; hence, . Further, we obtain . It is obvious that we have a minimum at this point in the interval .

This confirms the assertion.

If we put in the condition of the example, we receive . Therefore, , when , , , .

Theorem 1.4.

Let be a seminormed algebra over R with a unit. The functional satisfies condition (a): , for , as , where are nondecreasing functions such that .

Besides, the following requirements are fulfilled

Proof.

Here, we have , and , are nondecreasing.

Then, exist in compliance with (1). Therefore .

Let , , as . Let us put , where .

*.*

- (1)

Since does not decrease, and , then , where , , and .

We set , . As (2), we have , where is the rest of the sum.

Since does not decrease, and , then , where , , and . According to the induction principle, we obtain .

Corollary 1.5.

Let be a seminormed algebra above with a unit, and let be a seminormed space over , and .

Proof.

We consider the functional . Then, knowing (c), we get that satisfies Theorem 1.1's conditions and hence the needed inequality.

## Authors’ Affiliations

## References

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## Copyright

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