Open Access

Some New Results Related to Favard's Inequality

Journal of Inequalities and Applications20092009:128486

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

Received: 31 July 2008

Accepted: 5 February 2009

Published: 18 February 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 ,
(1.1)
and reverse inequality holds for . In [3], Simic considered the difference
(1.2)

He has given the following.

Theorem 1.1.

Let and be nonnegative and integrable functions on , with , then for , one has
(1.3)

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
(1.4)

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
(1.5)
Then
(1.6)

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
(1.7)

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
(1.8)

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
(1.9)

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
(1.10)

holds for each real and

Throughout the paper, we will frequently use the following family of convex functions on :
(1.11)

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
(1.12)

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
(2.1)
Then is log-convex for , and the following inequality holds for :
(2.2)

Proof.

Let us consider the function defined by
(2.3)
where , is defined by (1.11), and . We have
(2.4)
Therefore, is convex for . Using Theorem 1.4,
(2.5)
or equivalently
(2.6)
Since
(2.7)
we have
(2.8)
By Lemma 1.8, we have
(2.9)

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

Note that is continuous for since
(2.10)

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
(2.11)

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
(2.12)

Proof.

An equivalent form of (1.12) is
(2.13)
where , , , and . Since by Theorem 2.1, is log-convex, we can set in (2.13): , , , , and . We get
(2.14)

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
(2.15)
Then is log-convex for and the following inequality holds for :
(2.16)

Theorem 2.4.

Let , be defined as in Theorem 2.3, and let be nonnegative real numbers such that and one has
(2.17)

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
(3.1)
Then
(3.2)

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
(3.3)
Then
(3.4)

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
(3.5)
Then is log-convex on , and the following inequality holds for :
(3.6)
Let be an increasing convex function on , , . Then is log-convex on , and the following inequality holds for :
(3.7)

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
(3.8)
Let and be defined as in Theorem 3.2(2), and let be such that , , , and . Then,
(3.9)

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
(3.10)
Then is log-convex on , and the following inequality holds for :
(3.11)
Let be a decreasing convex function on , , . Then is log-convex on , and the following inequality holds for :
(3.12)

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
(3.13)
Let and be defined as in Theorem 3.4(2), and let be such that , , , and . Then
(3.14)

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
(3.15)

with zero for the function .

If is a positive decreasing concave function on , then Theorem 3.4 gives that is log-convex, and
(3.16)

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
(4.1)

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
(4.2)

Proof.

Set , . Applying (3.2) for and defined in Lemma 4.1, we have
(4.3)
that is,
(4.4)
(4.5)

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
(4.6)

provided that for every .

Proof.

Define the functional with
(4.7)
and set . Obviously, . Using Theorem 4.2 , there exists such that
(4.8)
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
(4.9)
Since is positive concave function, it is easy to see that is decreasing function on (see [6]), thus
(4.10)
so for every . Set
(4.11)
Obviously, , . By (4.9), obvious estimations and integration by parts, we have
(4.12)

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
(4.13)

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
(4.14)
In fact, similar result can also be given for (4.6). Namely, suppose that has inverse function. Then from (4.6), we have
(4.15)

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
(4.16)

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

Denote, and So by limit, we have
(4.17)

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

Theorem 4.6.

Let , then the following inequality is valid:
(4.18)

Proof.

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

Remark 4.7.

If , then the above means become
(4.19)

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

Declarations

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.

Authors’ Affiliations

(1)
Abdus Salam School of Mathematical Sciences, GC University
(2)
Faculty of Textile Technology, University of Zagreb
(3)
Faculty of Food Technology and Biotechnology, University of Zagreb

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Copyright

© Naveed Latif et al. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.