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.
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.
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.
We quote here another useful lemma from log-convexity theory (cf. ).
The following lemma is equivalent to the definition of convex function (see [4, page 2]).
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 .
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 .
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.
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.
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.
So, we have that the expression on the right-hand side of (4.15) is also a mean.
In our next result, we prove that this new mean is monotonic.
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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