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# Some New Results Related to Favard's Inequality

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 128486 (2009)

## Abstract

Log-convexity of Favard's difference is proved, and Drescher's and Lyapunov's type inequalities for this difference are deduced. The weighted case is also considered. Related Cauchy type means are defined, and some basic properties are given.

## 1. Introduction and Preliminaries

Let and be two positive measurable real valued functions defined on with . From theory of convex means (cf. [1, 2]), the well-known Jensen's inequality gives that for or ,

and reverse inequality holds for . In [3], Simic considered the difference

He has given the following.

Theorem 1.1.

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

Remark 1.2.

For an extension of Theorem 1.1 see [3].

Let us write the well-known Favard's inequality.

Theorem 1.3.

Let f be a concave nonnegative function on . If , then

If , the reverse inequality holds in (1.4).

Note that (1.4) is a reversion of (1.1) in the case when .

Let us note that Theorem 1.3 can be obtained from the following result and also obtained by Favard (cf. [4, page 212]).

Theorem 1.4.

Let be a nonnegative continuous concave function on , not identically zero, and let be a convex function on , where

Then

Karlin and Studden (cf. [5, page 412]) gave a more general inequality as follows.

Theorem 1.5.

Let be a nonnegative continuous concave function on , not identically zero; is defined in (1.5), and let be a convex function on , where c satisfies (where is the minimum of ). Then

For , , we can get the following from Theorem 1.5.

Theorem 1.6.

Let be continuous concave function such that ; is defined in (1.5). If , then

If , the reverse inequality holds in (1.8).

In this paper, we give a related results to (1.3) for Favard's inequality (1.4) and (1.8).

We need the following definitions and lemmas.

Definition 1.7.

It is said that a positive function is log-convex in the Jensen sense on some interval if

holds for every .

We quote here another useful lemma from log-convexity theory (cf. [3]).

Lemma 1.8.

A positive function f is log-convex in the Jensen sense on an interval if and only if the relation

holds for each real and

Throughout the paper, we will frequently use the following family of convex functions on :

The following lemma is equivalent to the definition of convex function (see [4, page 2]).

Lemma 1.9.

If is convex on an interval , then

holds for every .

Now, we will give our main results.

## 2. Favard's Inequality

In the following theorem, we construct another interesting family of functions satisfying the Lyapunov inequality. The proof is motivated by [3].

Theorem 2.1.

Let be a positive continuous concave function on ; is defined in (1.5), and

Then is log-convex for , and the following inequality holds for :

Proof.

Let us consider the function defined by

where , is defined by (1.11), and . We have

Therefore, is convex for . Using Theorem 1.4,

or equivalently

Since

we have

By Lemma 1.8, we have

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

Note that is continuous for since

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

Since is log-convex, that is, is convex, by Lemma 1.9 for and taking , we get

which is equivalent to (2.2).

Theorem 2.2.

Let , be defined as in Theorem 2.1, and let be nonnegative real numbers such that , , , and . Then

Proof.

An equivalent form of (1.12) is

where , , , and . Since by Theorem 2.1, is log-convex, we can set in (2.13): , , , , and . We get

from which (2.12) trivially follows.

The following extensions of Theorems 2.1 and 2.2 can be deduced in the same way from Theorem 1.5.

Theorem 2.3.

Let be a continuous concave function on such that ; is defined in (1.5), and

Then is log-convex for and the following inequality holds for :

Theorem 2.4.

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

## 3. Weighted Favard's Inequality

The weighted version of Favard's inequality was obtained by Maligranda et al. in [6].

Theorem 3.1.

Let be a positive increasing concave function on . Assume that is a convex function on , where

Then

If is an increasing convex function on and , then the reverse inequality in (3.2) holds.

Let be a positive decreasing concave function on . Assume that is a convex function on , where

Then

If is a decreasing convex function on and , then the reverse inequality in (3.4) holds.

Theorem 3.2.

Let be a positive increasing concave function on ; is defined in (3.1), and

Then is log-convex on , and the following inequality holds for :

Let be an increasing convex function on , , . Then is log-convex on , and the following inequality holds for :

Proof.

As in the proof of Theorem 2.1, we use Theorem 3.1(1) instead of Theorem 1.4.

Theorem 3.3.

Let and be defined as in Theorem 3.2(1), and let be such that , , , and . Then

Let and be defined as in Theorem 3.2(2), and let be such that , , , and . Then,

Proof.

Similar to the proof of Theorem 2.2.

Theorem 3.4.

Let be a positive decreasing concave function on ; is defined as in (3.3), and

Then is log-convex on , and the following inequality holds for :

Let be a decreasing convex function on , , . Then is log-convex on , and the following inequality holds for :

Proof.

As in the proof of Theorem 2.1, we use Theorem 3.1(2) instead of Theorem 1.4.

Theorem 3.5.

Let and be defined as in Theorem 3.4(1), and let be such that , , , and . Then

Let and be defined as in Theorem 3.4(2), and let be such that , , , and . Then

Proof.

Similar to the proof of Theorem 2.2.

Remark 3.6.

Let . If is a positive concave function on , then the decreasing rearrangement is concave on . By applying Theorem 3.4 to , we obtain that is log-convex. Equimeasurability of with gives and we see that Theorem 3.4 is equivalent to Theorem 2.1.

Remark 3.7.

Let with . Then Theorem 3.2 gives that if is a positive increasing concave function on , then is log-convex, and

with zero for the function .

If is a positive decreasing concave function on , then Theorem 3.4 gives that is log-convex, and

with zero for the function , where is the beta function, and is the harmonic number defined for with , where is the digamma function and the Euler constant.

## 4. Cauchy Means

Let us note that (2.12), (2.17), (3.8), (3.9), (3.13), and (3.14) have the form of some known inequalities between means (e.g., Stolarsky means, Gini means, etc.). Here we will prove that expressions on both sides of (3.8)) are also means. The proofs in the remaining cases are analogous.

Lemma 4.1.

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

are convex functions.

Theorem 4.2.

Let be a nonnegative integrable function on with . Let be a positive increasing concave function on , . Then there exists , such that

Proof.

Set , . Applying (3.2) for and defined in Lemma 4.1, we have

that is,

By combining (4.4) and (4.5), (4.2) follows from continuity of .

Theorem 4.3.

Let be a positive increasing concave nonlinear function on , and let be a nonnegative integrable function on with . If , then there exists such that

provided that for every .

Proof.

Define the functional with

and set . Obviously, . Using Theorem 4.2 , there exists such that

We give a proof that the expression in square brackets in (4.8) is nonzero (actually strictly positive by inequality (3.2)) for nonlinear function . Suppose that the expression in square brackets in (4.8) is equal to zero, which is by simple rearrangements equivalent to equality

Since is positive concave function, it is easy to see that is decreasing function on (see [6]), thus

so for every . Set

Obviously, , . By (4.9), obvious estimations and integration by parts, we have

This implies , which is equivalent to . This gives that is a linear function, which obviously implies that is a linear function.

Since the function is nonlinear, the expression in square brackets in (4.8) is strictly positive which implies that , and this gives (4.6). Notice that Theorem 4.2 for implies that the denominator of the right-hand side of (4.6) is nonzero.

Corollary 4.4.

Let be a nonnegative integrable function with . If is a positive increasing concave nonlinear function on , then for there exists such that

Proof.

Set and in (4.6), then we get (4.13).

Remark 4.5.

Since the function is invertible, then from (4.13) we have

In fact, similar result can also be given for (4.6). Namely, suppose that has inverse function. Then from (4.6), we have

So, we have that the expression on the right-hand side of (4.15) is also a mean.

By the inequality (4.14), we can consider

for as means in broader sense. Moreover, we can extend these means in other cases.

Denote, and So by limit, we have

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

Theorem 4.6.

Let , then the following inequality is valid:

Proof.

Since is log-convex, therefore by (3.8)) we get (4.18).

Remark 4.7.

If , then the above means become

In this way (4.18) for gives an extension of (2.12) (see Remark 3.6).

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## Acknowledgments

This research work is funded by Higher Education Commission Pakistan. The researches of the second author and third author are supported by the Croatian Ministry of Science, Education and Sports under the Research Grants 117-1170889-0888 and 058-1170889-1050, respectively.

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Latif, N., Pečarić, J. & Perić, I. Some New Results Related to Favard's Inequality.
*J Inequal Appl* **2009**, 128486 (2009). https://doi.org/10.1155/2009/128486

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DOI: https://doi.org/10.1155/2009/128486