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
and vice versa for .
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].
If is an increasing function, then
Let us note that if is a strictly increasing function, then equality in (1.7) is valid if we have equalities in (1.6) instead of inequalities, that is, and
The following similar result is also valid [1, page 153].
Let be an increasing function. If and if the following hold.
- (i)there exists an such that(1.8)
- (ii)If there exists an such that(1.9)
then the reverse of inequality in (1.7) holds.
2. Main Results
Then is a strictly increasing function for .
Since for , therefore is a strictly increasing function for .
Lemma 2.2 ().
holds for each real and .
The following lemma is equivalent to the definition of convex function (see [1, page 2]).
that is, is a positive-valued function.
This implies that is monotonically increasing.
Now by Lemma 2.2, we have that is log-convex in Jensen sense.
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.
Let and be two positive -tuples such that , all 's are not equal and
- (i)if such that condition (1.8) is satisfied, then is log-convex. Also for , we have(2.11)
- (ii)moreover if and (1.9) is satisfied, then we have that is log-convex and(2.12)
We will also use the following lemma.
from which (2.13) immediately follows.
Let us introduce the following.
Let us note that , and .
By raising power , we get (2.17) for .
From Remark 2.8, we get (2.17) is also valid for or or or .
Taking in (2.17), we get (2.22).
3. Mean Value Theorems
Then for are monotonically increasing functions.
that is, for are monotonically increasing functions.
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.
Then by setting , we get (3.16).
If and are pairwise distinct, then we put , and in (3.16) to get (3.18).
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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