On Logarithmic Convexity for Power Sums and Related Results II
© J. Pečarić and A. ur Rehman. 2008
Received: 14 October 2008
Accepted: 4 December 2008
Published: 16 December 2008
In the paper "On logarithmic convexity for power sums and related results" (2008), we introduced means by using power sums and increasing function. In this paper, we will define new means of convex type in connection to power sums. Also we give integral analogs of new means.
1. Introduction and Preliminaries
We introduced the Cauchy means involving power sums. Namely, the following results were obtained in .
We defined the following means.
In this paper, we introduce new Cauchy means of convex type in connection with Power sums. For means, we shall use the following result [1, page 154].
In Theorem 1.2, if is strictly convex, then (1.7) is strict unless and .
2. Discrete Result
where . Then is strictly convex for .
Here, we use the notation .
Since for , therefore is strictly convex for .
Lemma 2.2 (see ).
holds for each real and .
The following lemma is equivalent to definition of convex function [1, page 2].
By using the above lemmas and Theorem 1.2, as in , we can prove the following results.
Moreover, we can use (2.7) to obtain new means of Cauchy type involving power sums.
Let us introduce the following means.
Let us note that , and .
This implies that (1.4), which we derived in , is better than (2.7).
Let us note that there are not integral analogs of results from . Moreover, in Section 3 we will show that previous results have their integral analogs.
3. Integral Results
The following theorem is very useful for further result [1, page 159].
- (b)If and either there exists an such that(3.4)
then for every convex function such that for all , the reverse of the inequality in (3.3) holds.
To define the new means of Cauchy involving integrals, we define the following function.
This implies that is convex.
Now, by Lemma 2.2, we have is log-convex in Jensen sense.
Since , this implies that is continuous for all , therefore it is a log-convex [1, page 6].
which is equivalent to (3.7).
Let us note that , and .
By raising power , we get an inequality (3.13) for .
From Remark 3.6 , we get (3.13) is also valid for or or or .
Then for are convex.
that is, for are convex.
and (3.24) implies (3.20).
Moreover, (3.21) is valid if is bounded from above and again we have (3.20) is valid.
Of course (3.20) is obvious if is not bounded from above and below as well.
provided that denominators are nonzero.
After putting values, we get (3.25).
is valid, provided that all denominators are nonzero.
Then by setting , we get (3.32).
If and are pairwise distinct, then we put , and in (3.32) to get (3.34).
For other cases, we can consider limit as in Remark 3.6.
This research was partially funded by Higher Education Commission, Pakistan. The research of the first author was supported by the Croatian Ministry of Science, Education and Sports under the research Grant 117-1170889-0888.
- Pečarić J, 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.MATHGoogle Scholar
- Pečarić J, Rehman AU: On logarithmic convexity for power sums and related results. Journal of Inequalities and Applications 2008, 2008:-9.Google Scholar
- Simic S: On logarithmic convexity for differences of power means. Journal of Inequalities and Applications 2007, 2007:-8.Google Scholar
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