Cauchy Means of the Popoviciu Type

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 ,

(1.1)

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

(1.2)

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

(1.3)

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

(1.4)

where

(1.5)

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

(1.6)

Theorem 1.4.

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

Then

(1.7)

(1.8)

where

(1.9)

(1.10)

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

(1.11)

and equality in (1.8) holds if

(1.12)

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

Lemma 1.5.

Define the function

(1.13)

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

(1.14)

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 ,

(1.15)

if and only if the relation

(1.16)

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

(1.17)

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

(2.1)

and let be positive.

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

(2.2)

Proof.

Consider the function defined by

(2.3)

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

(2.4)

Therefore, is convex for . Using Theorem 1.3,

(2.5)

since

(2.6)

we have

(2.7)

By Lemma 1.7, we have

(2.8)

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

(2.9)

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

(2.10)

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

(2.11)

Proof.

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

(2.12)

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

and , , , . We get

(2.13)

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

(2.14)

where

(2.15)

for

(2.16)

and let be positive.

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

(2.17)

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

(2.18)

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

(2.19)

Then

(2.20)

for

(2.21)

Proof.

First, we solve these three integrals

(2.22)

Take

(2.23)

Substitute

(A)

and limits, when then , when then . So,

(2.24)

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

(2.25)

Take second integral

(2.26)

using integration by parts, we have

(2.27)

Let

(2.28)

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

(2.29)

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

(2.30)

Now, take third integral

(2.31)

Using integration by parts, we get

(2.32)

Let

(2.33)

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

(2.34)

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

(2.35)

Let

(2.36)

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

(2.37)

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

(2.38)

Then

(2.39)

For ,

(2.40)

Using , we have

(2.41)

For ,

(2.42)

Using , we have

(2.43)

For ,

(2.44)

Using , we have

(2.45)

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 ,

(2.46)

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

(2.47)

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

(3.1)

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

(3.2)

Proof.

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

(3.3)

that is,

(3.4)

(3.5)

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

(3.6)

then there exists such that

(3.7)

Provided that denominators are non-zero.

Proof.

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

(3.8)

where and are defined as follows:

(3.9)

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

Let be defined as follows:

(3.10)

Obviously, we have

(3.11)

On the other hand, there is an such that

(3.12)

By using the linearity property of the operator

(3.13)

Now and , we have from the last equation

(3.14)

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

(3.15)

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

(3.16)

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

(3.17)

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:

(3.18)

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

(3.19)

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

(3.20)

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

Theorem 3.6.

Let , then the following inequality is valid

(3.21)

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.