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.
1. Introduction and Preliminaries
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.
2. Main Results
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.
3. Mean Value Theorems
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.
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