# Differences of Weighted Mixed Symmetric Means and Related Results

- KhuramAli Khan
^{1}Email author, - J Pečarić
^{1, 2}and - I Perić
^{3}

**2010**:289730

https://doi.org/10.1155/2010/289730

© Khuram Ali Khan et al. 2010

**Received: **22 June 2010

**Accepted: **13 October 2010

**Published: **18 October 2010

## Abstract

Some improvements of classical Jensen's inequality are used to define the weighted mixed symmetric means. Exponential convexity and mean value theorems are proved for the differences of these improved inequalities. Related Cauchy means are also defined, and their monotonicity is established as an application.

## Keywords

## 1. Introduction and Preliminary Results

We obtain the monotonicity of these means as a consequence of the following improvement of Jensen's inequality [1].

Theorem 1.1.

If is a concave function, then the inequality (1.4) is reversed.

Corollary 1.2.

Proof.

Let such that , if , then we set , in (1.4) and raising the power , we get (1.6). Similarly we set , in (1.4) and raising the power , we get (1.7).

When or , we get the required results by taking limit.

We obtain generalized means by setting , and applying to (1.3).

Corollary 1.3.

where is convex and is decreasing, or is concave and is increasing.

Remark 1.4.

In fact Corollaries 1.2 and 1.3 are weighted versions of results in [2].

The inequality of Popoviciu as given by Vasić and Stanković in [3] (see also [4, page 173]) can be written in the following form:

Theorem 1.5.

where is given by (1.3) for convex function .

Corollary 1.6.

Proof.

Let such that , if , then we set , in (1.11) to obtain (1.13) and we set , in (1.11) to obtain (1.14).

When or , we get the required results by taking limit.

Corollary 1.7.

The following result is valid [5, page 8].

Theorem 1.8.

If is a concave function then the inequality (1.16) is reversed.

Corollary 1.9.

Proof.

Let such that , if , then we set , in (1.16) and raising the power , we get (1.19). Similarly we set , in (1.16) and raising the power , we get (1.20).

When or , we get the required results by taking limit.

We obtain these generalized means by setting , and applying to (1.17).

Corollary 1.10.

where is convex and is decreasing, or is concave and is increasing.

The following result is given in [4, page 90].

Theorem 1.11.

Corollary 1.12.

Proof.

Let such that , if , then we set , in (1.24) and raising the power , we get (1.27). Similarly we set , in (1.25) and raising the power , we get (1.28).

When or , we get the required results by taking limit.

We obtain these generalized means be setting , and applying to (1.25).

Corollary 1.13.

where is convex and is decreasing, or is concave and is increasing.

The following result is given at [4, page 97].

Theorem 1.14.

Corollary 1.15.

Proof.

Let such that , if , then we set , in (1.32) and raising the power , we get (1.36). Similarly we set , in (1.32) and raising the power , we get (1.37).

When or , we get the required results by taking limit.

We obtain these generalized means by setting , and applying to (1.34).

Corollary 1.16.

where is convex and is decreasing, or is concave and is increasing.

Remark 1.17.

In fact unweighted version of these results were proved in [6], but in Remark 2.14 from [6], it is written that the same is valid for weighted case.

The exponentially convex functions are defined in [7] as follows.

Definition 1.18.

for all and all choices and , .

We also quote here a useful propositions from [7].

Proposition 1.19.

Let be a function, then following statements are equivalent;

Proposition 1.20.

If is an exponentially convex function then is a log-convex function.

It is easy to see that both and are convex.

In this paper we prove the exponential convexity of (1.43) for convex functions defined in (1.46) and (1.47) and mean value theorems for the differences given in (1.43). We also define the corresponding means of Cauchy type and establish their monotonicity.

## 2. Main Result

The following theorems are the generalizations of results given in [6].

where is obtained by replacing convex function with for , in . Then the following statements are valid.

(b)The function is exponentially convex on .

- (ii)
It was proved in [6] that is continuous for . By using Proposition 1.19, we get exponential convexity of the function .

Theorem 2.2.

Theorem 2.1 is still valid for convex functions .

Theorem 2.3.

Proof.

are convex.

Theorem 2.4.

provided that the denominators are non zero.

Proof.

Corollary 2.5.

Proof.

Taking and , in (2.13), for distinct real numbers and , different from zero and 1, we obtain (2.18).

Remark 2.6.

## 3. Cauchy Mean

The following lemma gives an equivalent definition of the convex function [4, page 2].

Lemma 3.1.

Now, we deduce the monotonicity of means given in (3.2) in the form of Dresher's inequality, as follows.

Theorem 3.2.

Proof.

This together with (2.1) follows (3.5).

Corollary 3.3.

Proof.

Set and , then taking in (2.13), we get (3.7) by the virtue of (1.2), (1.18), (1.26) and (1.35) for non zero, distinct real numbers , and .

Remark 3.4.

where , , and are non zero, distinct real numbers.

Now, we give the monotonicity of new means given in (3.10), as follows:

Theorem 3.5.

Proof.

For , we set , , , , such that , , in (2.1) to obtain (3.12) with the help of (3.13).

Similarly for , we set , , , , such that , , in (2.1) and get (3.12) again, by the virtue of (3.13).

In the case , since for is continuous therefore We get required result by taking limit.

## Declarations

### Acknowledgments

This research was partially funded by Higher Education Commission, Pakistan. The research of the second author was supported by the Croatian Ministry of Science, Education and Sports under the Research Grant no. 117-1170889-0888.

## Authors’ Affiliations

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## Copyright

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