# On Logarithmic Convexity for Power Sums and Related Results

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

**2008**:389410

**DOI: **10.1155/2008/389410

© J. Pečarić and A. U. Rehman. 2008

**Received: **28 March 2008

**Accepted: **29 June 2008

**Published: **7 July 2008

## Abstract

We give some further consideration about logarithmic convexity for differences of power sums inequality as well as related mean value theorems. Also we define quasiarithmetic sum and give some related results.

## 1. Introduction and Preliminaries

and vice versa for .

Simić [2] has considered the difference of both sides of (1.1). He considers the function defined as

and has proved the following theorem.

Theorem 1.1.

Anwar and Pečarić [3] have considerd further generalization of Theorem 1.1. Namely, they introduced new means of Cauchy type in [4] and further proved comparison theorem for these means.

In this paper, we will give some results in the case where instead of means we have power sums.

Let us note that (1.5) can also be obtained from the following theorem [1, page 152].

Theorem 1.2.

If is an increasing function, then

Remark 1.3.

Let us note that if is a strictly increasing function, then equality in (1.7) is valid if we have equalities in (1.6) instead of inequalities, that is, and

The following similar result is also valid [1, page 153].

Theorem 1.4.

Let be an increasing function. If and if the following hold.

- (i)there exists an such that(1.8)

- (ii)If there exists an such that(1.9)

then the reverse of inequality in (1.7) holds.

In this paper, we will give some applications of power sums. That is, we will prove results similar to those shown in [2, 3], but for power sums.

## 2. Main Results

Lemma 2.1.

Then is a strictly increasing function for .

Proof.

Since for , therefore is a strictly increasing function for .

Lemma 2.2 ([2]).

holds for each real and .

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

Lemma 2.3.

Theorem 2.4.

Proof.

that is, is a positive-valued function.

This implies that is monotonically increasing.

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

Since it follows that is continuous, therefore it is a log-convex function [1, page 6].

which is equivalent to (2.6).

Similar application of Theorem 1.4 gives the following.

Theorem 2.5.

Let and be two positive -tuples such that , all 's are not equal and

- (i)if such that condition (1.8) is satisfied, then is log-convex. Also for , we have(2.11)

- (ii)moreover if and (1.9) is satisfied, then we have that is log-convex and(2.12)

We will also use the following lemma.

Lemma 2.6.

Proof.

from which (2.13) immediately follows.

Let us introduce the following.

Definition 2.7.

Remark 2.8.

Let us note that , and .

Theorem 2.9.

Proof.

By raising power , we get (2.17) for .

From Remark 2.8, we get (2.17) is also valid for or or or .

Corollary 2.10.

Proof.

Taking in (2.17), we get (2.22).

## 3. Mean Value Theorems

Lemma 3.1.

Then for are monotonically increasing functions.

Proof.

that is, for are monotonically increasing functions.

Theorem 3.2.

Proof.

and (3.7) implies (3.4).

Theorem 3.3.

provided that the denominators are nonzero.

Proof.

After putting values, we get (3.8).

Theorem 3.4.

is valid, provided that all denominators are not zero.

Proof.

Then by setting , we get (3.16).

Corollary 3.5.

Proof.

If and are pairwise distinct, then we put , and in (3.16) to get (3.18).

For other cases, we can consider limit as in Remark (2.8).

## Declarations

### Acknowledgment

The authors are really very thankful to Mr. Martin J. Bohner for his useful suggestions.

## Authors’ Affiliations

## References

- Pečarić JE, 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 - Simić S:
**On logarithmic convexity for differences of power means.***Journal of Inequalities and Applications*2007,**2007:**-8.Google Scholar - Anwar M, Pečarić JE:
**On logarithmic convexity for differences of power means.**to appear in*Mathematical Inequalities & Applications* - Anwar M, Pečarić JE:
**New means of Cauchy's type.***Journal of Inequalities and Applications*2008,**2008:**-10.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.