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].
2. Discrete Result
Lemma 2.2 (see ).
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.
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].
To define the new means of Cauchy involving integrals, we define the following function.
Since , this implies that is continuous for all , therefore it is a log-convex [1, page 6].
which is equivalent to (3.7).
and (3.24) implies (3.20).
provided that denominators are nonzero.
After putting values, we get (3.25).
is valid, provided that all denominators are nonzero.
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.
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