# Cauchy Means of the Popoviciu Type

- Matloob Anwar
^{1}, - Naveed Latif
^{1}Email author and - J. Pečarić
^{1, 2}

**2009**:628051

https://doi.org/10.1155/2009/628051

© Matloob Anwar et al. 2009

**Received: **9 October 2008

**Accepted: **2 February 2009

**Published: **4 February 2009

## Abstract

We discuss log-convexity for the differences of the Popoviciu inequalities and introduce some mean value theorems and related results. Also we give the Cauchy means of the Popoviciu type and we show that these means are monotonic.

## 1. Introduction and Preliminaries

The following result was given in [4] (see also [5]).

Theorem 1.1.

Remark 1.2.

For extension of Theorem 1.1 see (cf. [4]).

Popoviciu ([6–8], [9, pages 214-215]) has proved the following results.

Theorem 1.3.

Theorem 1.4.

Let be continuous and convex, and let be convex of order such that for .

With the help of the following useful lemmas we prove our results.

Lemma 1.5.

Then , that is, is convex for .

The following lemma is equivalent to definition of convex function (see [9, page 2]).

Lemma 1.6.

We quote here another useful lemma from log-convexity theory (cf. [4]).

Lemma 1.7.

The following lemma given in [10] gives the relation between Beta function and Hypergeometric function .

Lemma 1.8.

The paper is organized in the following way. After this introduction, in the second section we discuss the log-convexity of differences of the Popoviciu inequalities (1.4), (1.7), and (1.8). In the third section we introduce some mean value theorems and the Cauchy means of the Popoviciu-type and discuss its monotonicity.

## 2. Main Results

Theorem 2.1.

Proof.

This implies is continuous, therefore it is log-convex.

which is equivalent to (2.2).

Theorem 2.2.

Proof.

Since by Theorem 2.1, is log-convex, we can set in (2.12):

and after applying exponential function, we get (2.11).

Theorem 2.3.

Proof.

As in the proof of Theorem 2.1, we use Theorem 1.4 instead of Theorem 1.3.

Theorem 2.4.

Proof.

Similar to the proof of Theorem 2.2.

Lemma 2.5.

Proof.

Theorem 2.6.

Let be convex of order such that for , be defined in (1.5) and let defined in (2.20) be positive.

Proof.

As in the proof of Theorem 2.1, we use Theorem 1.4 instead of Theorem 1.3.

Theorem 2.7.

Proof.

Similar to the proof of Theorem 2.2.

## 3. Cauchy Means

Let us note that (2.11) has the form of some known inequalities between means (e.g., Stolarsky's means, etc.). Here we prove that expressions on both sides of (2.11) are also means.

Lemma 3.1.

are convex functions.

Theorem 3.2.

Proof.

By combining (3.4) and (3.5) and using the fact that for there exists such that we get (3.2).

Theorem 3.3.

Provided that denominators are non-zero.

Proof.

Since and satisfy (3.2), therefore as linear combination of and should also satisfy (3.2).

After putting values, we get (3.7).

Corollary 3.4.

Proof.

Set and in (3.7) we get (3.15).

Remark 3.5.

The expression on the right-hand side of (3.17) is also a mean.

for . Moreover we can extend these means in other cases. By limit we have

In our next result we prove that this new mean is monotonic.

Theorem 3.6.

Proof.

Since is log-convex, therefore by (2.11) we get (3.21).

Remark 3.7.

Similar results of the Cauchy means and related results can also proved for Theorems 2.3 and 2.6.

## Declarations

### Acknowledgments

This research work is funded by Higher Education Commission Pakistan. The research of the third author is supported by the Croatian Ministry of Science, Education and Sports under the Research Grants 117-1170889-0888.

## Authors’ Affiliations

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