- Research Article
- Open Access
On Logarithmic Convexity for Power Sums and Related Results
© J. Pečarić and A. U. Rehman. 2008
- Received: 28 March 2008
- Accepted: 29 June 2008
- Published: 7 July 2008
We give some further consideration about logarithmic convexity for differences of power sums inequality as well as related mean value theorems. Also we define quasiarithmetic sum and give some related results.
- Continuous Function
- Real Number
- Convex Function
- Positive Function
- Related Result
Simić  has considered the difference of both sides of (1.1). He considers the function defined as
and has proved the following theorem.
In this paper, we will give some results in the case where instead of means we have power sums.
Let us note that (1.5) can also be obtained from the following theorem [1, page 152].
The following similar result is also valid [1, page 153].
then the reverse of inequality in (1.7) holds.
Lemma 2.2 ().
The following lemma is equivalent to the definition of convex function (see [1, page 2]).
Since it follows that is continuous, therefore it is a log-convex function [1, page 6].
which is equivalent to (2.6).
Similar application of Theorem 1.4 gives the following.
We will also use the following lemma.
from which (2.13) immediately follows.
Let us introduce the following.
and (3.7) implies (3.4).
provided that the denominators are nonzero.
After putting values, we get (3.8).
is valid, provided that all denominators are not zero.
For other cases, we can consider limit as in Remark (2.8).
The authors are really very thankful to Mr. Martin J. Bohner for his useful suggestions.
- 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.MATHGoogle Scholar
- Simić S: On logarithmic convexity for differences of power means. Journal of Inequalities and Applications 2007, 2007:-8.Google Scholar
- Anwar M, Pečarić JE: On logarithmic convexity for differences of power means. to appear in Mathematical Inequalities & ApplicationsGoogle Scholar
- Anwar M, Pečarić JE: New means of Cauchy's type. Journal of Inequalities and Applications 2008, 2008:-10.Google Scholar
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.