General Convexity of Some Functionals in Seminormed Spaces and Seminormed Algebras

Abstract

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

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,

(1)there exists , where

(1.1)

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

Proof.

Let , as . We put , where , , .

• (a) Let . According to condition (a), we obtain

  

  (1.2)

Knowing that and are nondecreasing, we obtain

(1.3)

where .

There exists in compliance with (1). Therefore .

If we put the result is , that is, .

• (b) Let . Then, in view of (a), we have

  

  (1.4)

Let us consider elements ,, and we suppose .

Let , where , and , , as .

According to condition (a), we get

(1.5)

where , .

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

We know that , and therefore about the statement is proved. We assume the assertion about is correct.

• (1) Let . Then, , where is the rest of the sum, and , . With condition (2) we have but . Setting and knowing is nondecreasing function, we obtain

  

  (1.6)

where and . With the inductive assumption, , that is, , that is, .

• (2) Let . Then , where is the rest of the sum, and , . According to condition (2), we obtain . Let us place, where , but and is a nondecreasing function. Then, , where , and .

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.

If we put in the conditions of Theorem 1.1, , , , , and , , , then about ,, , , we will obtain the inequality

(1.7)

where

(1.8)

Proof.

Let us consider , where

(1.9)

Then, ,

(1.10)

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

Then, we obtain , and hence at the same point

(1.11)

since

(1.12)

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

(1)There exists , where

(1.13)

(2)The function, and are convex. Then, if ,  , one receives the inequality

(1.14)

Proof.

Let , as , .

We put , where .

• (a) Let . According to condition (a), we have .

Here, we have , and , are nondecreasing.

If , , then , where .

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

If we put , the result is , that is, .

• (b) Let . Then, in view of the fact that (a), we get

  

  (1.15)

Let , , as . Let us put , where .

We can accept . Let and .

We have , where

(1.16)

Applying the principle of induction over we will prove that . In view of the fact that was mentioned at the beginning, we get . Assuming the statement for holds, we will prove it for .

• (1) Let .

Putting , , we have where is the rest of the sum. Using condition (2), we get

(1.17)

Let , .

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

By we denote the unit of the algebra . According to the inductive suggestion, we obtain .

• (2) Let .

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

Let , .

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 .

Then, if one replaces the condition (a) in Theorem 1.4 by condition (c): , for all with , and all the rest of the conditions are satisfied. One denotes by the norm in , and the norm in with . Then if , one receives the inequality

(1.18)

Proof.

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