Some New Results Related to Favard's Inequality
© Naveed Latif et al. 2009
Received: 31 July 2008
Accepted: 5 February 2009
Published: 18 February 2009
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
He has given the following.
For an extension of Theorem 1.1 see .
Let us write the well-known Favard's inequality.
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]).
Karlin and Studden (cf. [5, page 412]) gave a more general inequality as follows.
For , , we can get the following from Theorem 1.5.
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.
holds for every .
We quote here another useful lemma from log-convexity theory (cf. ).
holds for each real and
The following lemma is equivalent to the definition of convex function (see [4, page 2]).
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 .
that is, is log-convex in the Jensen sense for .
This implies is continuous; therefore, it is log-convex.
which is equivalent to (2.2).
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.
3. Weighted Favard's Inequality
The weighted version of Favard's inequality was obtained by Maligranda et al. in .
If is an increasing convex function on and , then the reverse inequality in (3.2) holds.
If is a decreasing convex function on and , then the reverse inequality in (3.4) holds.
As in the proof of Theorem 2.1, we use Theorem 3.1(1) instead of Theorem 1.4.
Similar to the proof of Theorem 2.2.
As in the proof of Theorem 2.1, we use Theorem 3.1(2) instead of Theorem 1.4.
Similar to the proof of Theorem 2.2.
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.
with zero for the function .
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.
are convex functions.
By combining (4.4) and (4.5), (4.2) follows from continuity of .
provided that for every .
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.
Set and in (4.6), then we get (4.13).
So, we have that the expression on the right-hand side of (4.15) is also a mean.
for as means in broader sense. Moreover, we can extend these means in other cases.
In our next result, we prove that this new mean is monotonic.
Since is log-convex, therefore by (3.8)) we get (4.18).
In this way (4.18) for gives an extension of (2.12) (see Remark 3.6).
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.
- Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge, UK; 1978.Google Scholar
- Mitrinović DS: Analytic Inequalities, Die Grundlehren der mathematischen Wissenschaften. Volume 165. Springer, Berlin, Germany; 1970:xii+400.Google Scholar
- Simic S: On logarithmic convexity for differences of power means. Journal of Inequalities and Applications 2007, 2007:-8.Google Scholar
- Pečarić JE, Proschan F, Tong YL: Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering. Volume 187. Academic Press, Boston, Mass, USA; 1992:xiv+467.Google Scholar
- Karlin S, Studden WJ: Tchebycheff Systems: With Applications in Analysis and Statistics, Pure and Applied Mathematics. Volume 15. Interscience, New York, NY, USA; 1966:xviii+586.Google Scholar
- Maligranda L, Pečarić JE, Persson LE: Weighted Favard and Berwald inequalities. Journal of Mathematical Analysis and Applications 1995,190(1):248–262. 10.1006/jmaa.1995.1075MathSciNetView ArticleMATHGoogle Scholar
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