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# Cauchy Means of the Popoviciu Type

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 628051 (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

Let and be two positive real valued functions with , then from theory of convex means (cf. [1–3]), the well-known Jensen inequality gives that for or ,

and vise versa for . In [4], Simic has considered the difference

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

Theorem 1.1.

Let , be nonnegative and integrable functions for , with , then for ; , one has

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.

Let be convex and be continuous, increasing, and convex such that for . Then

where

If is strictly convex, then the equality in (1.4) holds if and only if

Theorem 1.4.

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

Then

where

If is strictly convex, then equality in (1.7) holds if and only if

and equality in (1.8) holds if

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

Lemma 1.5.

Define the function

Then , that is, is convex for .

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

Lemma 1.6.

If is a convex function on for all for which , the following is valid

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

Lemma 1.7.

A positive function f is log-convex in the Jensen-sense on an open interval , that is, for each ,

if and only if the relation

holds for each real and

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

Lemma 1.8.

Suppose are such that and , and are Beta and Hypergeometric functions, respectively. Then

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.

Let be continuous, increasing, and convex such that for , and let be defined in (1.5) and

and let be positive.

One has that is log-convex and the following inequality holds for ,

Proof.

Consider the function defined by

where , is defined by (1.13) and . We have

Therefore, is convex for . Using Theorem 1.3,

since

we have

By Lemma 1.7, we have

that is is log-convex in the Jensen-sense for . Since

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

Since is log-convex, that is, is convex, therefore by Lemma 1.6 for and taking , we get

which is equivalent to (2.2).

Theorem 2.2.

Let , be defined in Theorem 2.1 and let be real numbers such that , , , , one has

Proof.

In (cf. [9, page 2]), we have the following result for convex function with , , , :

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

and , , , . We get

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

Theorem 2.3.

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

where

for

and let be positive.

One has that is log-convex and the following inequality holds for ,

Proof.

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

Theorem 2.4.

Let , be defined in Theorem 2.3 and be real numbers such that , , , , one has

Proof.

Similar to the proof of Theorem 2.2.

Lemma 2.5.

Let be convex of order such that for , be defined in (1.5) and be defined in (1.10), and let and are Beta and Hypergeometric functions respectively, and

Then

for

Proof.

First, we solve these three integrals

Take

Substitute

and limits, when then , when then . So,

By using Lemma 1.8 with , , , , such that and , we get

Take second integral

using integration by parts, we have

Let

By using same substitution (A) as above, we get

By using Lemma 1.8 with , , , , such that and , we get

Now, take third integral

Using integration by parts, we get

Let

By using same substitution (A) as above, we get

By using Lemma 1.8 with , , , , such that and , we get

Let

By using same substitution (A) as above, we have

By using Lemma 1.8 with , , , , such that and , we get

Then

For ,

Using , we have

For ,

Using , we have

For ,

Using , we have

Theorem 2.6.

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

One has that is log-convex and the following inequality holds for ,

Proof.

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

Theorem 2.7.

Let , be defined in Theorem 2.6 and be real numbers such that , , , , one has

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.

Let be such that is bounded, that is, Then the functions defined by

are convex functions.

Theorem 3.2.

Let is a compact interval in and be a continuous, increasing and convex such that for , be defined in (1.5) then , such that

Proof.

Suppose for . Then by applying and defined in Lemma 3.1 for in (1.4), we have

that is,

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

Theorem 3.3.

Let and satisfy (3.2), f be a continuous, increasing and convex such that for , be defined in (1.5), and

then there exists such that

Provided that denominators are non-zero.

Proof.

Consider the linear functionals and such that for some function and . Consider the following linear combination

where and are defined as follows:

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

Let be defined as follows:

Obviously, we have

On the other hand, there is an such that

By using the linearity property of the operator

Now and , we have from the last equation

After putting values, we get (3.7).

Corollary 3.4.

Let f be a continuous, increasing and convex such that for , then for there exists such that

where and .

Proof.

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

Remark 3.5.

Since the function is invertible, therefore from (3.15) we have

In fact, similar result can also be given for (3.7). Namely, suppose that has inverse function. Then from (3.7) we have

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

From the inequality (3.16), we can define means as follows:

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

where denotes , denotes , denotes , denotes , denotes , and denotes , where

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

Theorem 3.6.

Let , then the following inequality is valid

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.

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## 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.

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Anwar, M., Latif, N. & Pečarić, J. Cauchy Means of the Popoviciu Type.
*J Inequal Appl* **2009, **628051 (2009). https://doi.org/10.1155/2009/628051

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### Keywords

- Linear Combination
- Real Number
- Exponential Function
- Convex Function
- Integrable Function