# On Logarithmic Convexity for Power Sums and Related Results II

- J. Pečarić
^{1, 2}and - Atiq ur Rehman
^{1}Email author

**2008**:305623

**DOI: **10.1155/2008/305623

© J. Pečarić and A. ur Rehman. 2008

**Received: **14 October 2008

**Accepted: **4 December 2008

**Published: **16 December 2008

## Abstract

In the paper "On logarithmic convexity for power sums and related results" (2008), we introduced means by using power sums and increasing function. In this paper, we will define new means of convex type in connection to power sums. Also we give integral analogs of new means.

## 1. Introduction and Preliminaries

**x**be positive

*n*-tuples. The well-known inequality for power sums of order and , for (see [1, page 164]), states that

We introduced the Cauchy means involving power sums. Namely, the following results were obtained in [2].

We defined the following means.

Definition 1.1.

**x**and

**p**be two nonnegative

*n*-tuples such that . Then for ,

In this paper, we introduce new Cauchy means of convex type in connection with Power sums. For means, we shall use the following result [1, page 154].

Theorem 1.2.

*n*-tuples such that condition (1.5) is valid. If is a convex function on , then

Remark 1.3.

In Theorem 1.2, if is strictly convex, then (1.7) is strict unless and .

## 2. Discrete Result

Lemma 2.1.

where . Then is strictly convex for .

Here, we use the notation .

Proof.

Since for , therefore is strictly convex for .

Lemma 2.2 (see [3]).

holds for each real and .

The following lemma is equivalent to definition of convex function [1, page 2].

Lemma 2.3.

Lemma 2.4.

By using the above lemmas and Theorem 1.2, as in [2], we can prove the following results.

Theorem 2.5.

Moreover, we can use (2.7) to obtain new means of Cauchy type involving power sums.

Let us introduce the following means.

Definition 2.6.

**x**and

**p**be two nonnegative

*n*-tuples such that then for ,

Remark 2.7.

Let us note that , and .

Theorem 2.8.

Theorem 2.9.

Remark 2.10.

This implies that (1.4), which we derived in [2], is better than (2.7).

Let us note that there are not integral analogs of results from [2]. Moreover, in Section 3 we will show that previous results have their integral analogs.

## 3. Integral Results

The following theorem is very useful for further result [1, page 159].

Theorem 3.1.

- (a)If(3.2)

- (b)If and either there exists an such that(3.4)

then for every convex function such that for all , the reverse of the inequality in (3.3) holds.

To define the new means of Cauchy involving integrals, we define the following function.

Definition 3.2.

Theorem 3.3.

Proof.

This implies that is convex.

Now, by Lemma 2.2, we have is log-convex in Jensen sense.

Since , this implies that is continuous for all , therefore it is a log-convex [1, page 6].

which is equivalent to (3.7).

Theorem 3.4.

Definition 3.5.

Remark 3.6.

Let us note that , and .

Theorem 3.7.

Proof.

By raising power , we get an inequality (3.13) for .

From Remark 3.6 , we get (3.13) is also valid for or or or .

Lemma 3.8.

Then for are convex.

Proof.

that is, for are convex.

Theorem 3.9.

Proof.

and (3.24) implies (3.20).

Moreover, (3.21) is valid if is bounded from above and again we have (3.20) is valid.

Of course (3.20) is obvious if is not bounded from above and below as well.

Theorem 3.10.

provided that denominators are nonzero.

Proof.

After putting values, we get (3.25).

Theorem 3.11.

is valid, provided that all denominators are nonzero.

Proof.

Then by setting , we get (3.32).

Corollary 3.12.

Proof.

If and are pairwise distinct, then we put , and in (3.32) to get (3.34).

For other cases, we can consider limit as in Remark 3.6.

## Declarations

### Acknowledgments

This research was partially funded by Higher Education Commission, Pakistan. The research of the first author was supported by the Croatian Ministry of Science, Education and Sports under the research Grant 117-1170889-0888.

## Authors’ Affiliations

## References

- Pečarić J, Proschan F, Tong YL:
*Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering*.*Volume 187*. Academic Press, Boston, Mass, USA; 1992:xiv+467.MATHGoogle Scholar - Pečarić J, Rehman AU:
**On logarithmic convexity for power sums and related results.***Journal of Inequalities and Applications*2008,**2008:**-9.Google Scholar - Simic S:
**On logarithmic convexity for differences of power means.***Journal of Inequalities and Applications*2007,**2007:**-8.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.