# Generalization of Stolarsky Type Means

- J Pečarić
^{1, 2}and - G Roqia
^{2}Email author

**2010**:720615

https://doi.org/10.1155/2010/720615

© J. Pečarić and G. Roqia. 2010

**Received: **27 April 2010

**Accepted: **15 October 2010

**Published: **20 October 2010

## Abstract

We generalize means of Stolarsky type and show the monotonicity of these generalized means.

## Keywords

## 1. Introduction and Preliminaries

provided that is a convex function [1, page 137], [2, page 1].

This result for convex functions plays an important role in nonlinear analysis. These classical inequalities have been improved and generalized in a number of ways and applied for special means including Stolarsky type, logarithmic, and -logarithmic means. A generalization of H.H inequalities was obtained in [3–5], [2, page 5], and [1, page 143].

Theorem 1.1.

Remark 1.2.

The inequalities given by (1.2) are strict if is a continuous strictly convex on .

The above inequality is strict, when is strictly convex continuous function.

Remark 1.3.

Some simple proofs of inequality (1.8) and related results on means of Stolarsky type are given in [6].

The aim of this paper is to prove the exponential convexity of the functions deduced from (1.5) and apply these functions to generalize the means of Stolarsky type, and at last we prove the monotonicity property of these new means.

We review some necessary definitions and preliminary results.

Definition 1.4 (see [7]).

for each and every , such that , , .

Proposition 1.5 (see [7]).

Definition 1.6 (see [1]).

Corollary 1.7 (see [7]).

If is exponentially convex then is log-convex function.

The following lemma is another way to define convex function [1, page 2].

Lemma 1.8.

In Section 2, we prove the exponential and logarithmic convexity of the functions deduced from (1.5). We also prove related mean value theorems of Cauchy type.

## 2. Main Results

The following lemma gives us very important family of convex functions.

Lemma 2.1 (see [7]).

Theorem 2.2.

where is defined in Lemma 2.1. Then

matrix is positive semidefinite for each and ; particularly,

the function is exponentially convex on ;

if , then the function is a log-convex on and the following inequality holds for such that ;

Proof.

(i)Consider the function

(ii)Since for , it follows that is continuous on . Therefore, by Proposition 1.5 for , we get exponential convexity of on .

(iii)Let , then the log-convexity of is a simple consequence of Corollary 1.7. By setting , , , in Lemma 1.8, we have

which implies (2.4).

We will use the following lemma in the proof of mean value theorem.

Lemma 2.3 (see [1, page 4]).

Proof.

Theorem 2.4.

Proof.

We get the required result.

Theorem 2.5.

Proof.

which is clearly (2.20).

Corollary 2.6.

Remark 2.7.

## 3. Means of Stolarsky Type

for . We will use the following lemma to prove the monotonicity of Stolarsky type means.

Lemma 3.1.

The proof of this Lemma is given in [1].

Theorem 3.2.

Proof.

which is equivalent to (3.5) for , . By continuity of , (3.5) is valid for , .

Remark 3.3.

If we substitute and replace and in , for , then means of Stolarsky type and related results given in [6] are obtained.

## 4. Generalized Means of Stolarsky Type

Theorem 4.1.

Theorem 2.2 is still valid if one sets .

Proof.

The proof is similar to the proof of Theorem 2.2.

Theorem 4.2.

Proof.

By raising power , to (4.13) and , to (4.14), we get (4.11) for , .

which completes the proof.

Remark 4.3.

If we substitute , , and in the above results, then the results of generalized Stolarsky type means proved in [6] are recaptured.

## Declarations

### Acknowledgment

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

## 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.Google Scholar - Dragomir SS, Pearce CEM:
*Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monograph Collections*. Victoria University; 2002.Google Scholar - Vasić PM, Lacković IB: On an inequality for convex functions.
*Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika*1974, (461–497):63–66.Google Scholar - Lupaş A: A generalization of Hadamard inequalities for convex functions.
*Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika*1976, (544–576):115–121.Google Scholar - Vasić PM, Lacković IB: Some complements to the paper: "On an inequality for convex functions".
*Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika*1976, (544–576):59–62.Google Scholar - Jakšetić J, Pečarić J, ur Rehman A: On Stolarsky and related means.
*Mathematical Inequalities and Applications*2010, 13: 899–909.MathSciNetGoogle Scholar - Anwar M, Jakšetić J, Pečarić J, ur Rehman A: Exponential convexity, positive semi-definite matrices and fundamental inequalities.
*Journal of Mathematical Inequalities*2010, 4(2):171–189.MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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