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On Logarithmic Convexity for Power Sums and Related Results
Journal of Inequalities and Applications volume 2008, Article number: 389410 (2008)
Abstract
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
Let , denote two sequences of positive real numbers with . The wellknown Jensen Inequality [1, page 43] gives the following, for or :
and vice versa for .
Simić [2] has considered the difference of both sides of (1.1). He considers the function defined as
and has proved the following theorem.
Theorem 1.1.
For , then
Anwar and Pečarić [3] have considerd further generalization of Theorem 1.1. Namely, they introduced new means of Cauchy type in [4] and further proved comparison theorem for these means.
In this paper, we will give some results in the case where instead of means we have power sums.
Let be positive tuples. The wellknown inequality for power sums of order and , for (see [1, page 164]), states that
Moreover, if is a positive tuples such that , then for (see [1, page 165]), we have
Let us note that (1.5) can also be obtained from the following theorem [1, page 152].
Theorem 1.2.
Let and be two nonnegative tuples such that and
If is an increasing function, then
Remark 1.3.
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].
Theorem 1.4.
Let be an increasing function. If and if the following hold.

(i)
there exists an such that
(1.8)
where and , then (1.7) holds.

(ii)
If there exists an such that
(1.9)
then the reverse of inequality in (1.7) holds.
In this paper, we will give some applications of power sums. That is, we will prove results similar to those shown in [2, 3], but for power sums.
2. Main Results
Lemma 2.1.
Let
Then is a strictly increasing function for .
Proof.
Since for , therefore is a strictly increasing function for .
Lemma 2.2 ([2]).
A positive function is log convex in Jensen's sense on an open interval , that is, for each ,
if and only if the relation
holds for each real and .
The following lemma is equivalent to the definition of convex function (see [1, page 2]).
Lemma 2.3.
If is continuous and convex for all , , of an open interval for which , then
Theorem 2.4.
Let and be two positive tuples and let
such that condition (1.6) is satisfied and all 's are not equal. Then is logconvex, also for where we have
Proof.
Since is a strictly increasing function for and all 's are not equal, therefore by Theorem 1.2 with , we have
that is, is a positivevalued function.
Let where and :
This implies that is monotonically increasing.
By Theorem 1.2, we have
Now by Lemma 2.2, we have that is logconvex in Jensen sense.
Since it follows that is continuous, therefore it is a logconvex function [1, page 6].
Since is logconvex, that is, is convex, we have by Lemma 2.3 that, for with ,
which is equivalent to (2.6).
Similar application of Theorem 1.4 gives the following.
Theorem 2.5.
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 logconvex. Also for , we have
(2.11)

(ii)
moreover if and (1.9) is satisfied, then we have that is logconvex and
(2.12)
We will also use the following lemma.
Lemma 2.6.
Let f be a logconvex function and assume that if . Then the following inequality is valid:
Proof.
In [1, page 2], we have the following result for convex function , with :
Putting , we get
from which (2.13) immediately follows.
Let us introduce the following.
Definition 2.7.
Let and be two nonnegative tuples such that then for , we define
Remark 2.8.
Let us note that , and .
Theorem 2.9.
Let such that . Then we have
Proof.
Let
Now taking , where , and in Lemma 2.6, we have
Since by substituting and , where , in above inequality, we get
By raising power , we get (2.17) for .
From Remark 2.8, we get (2.17) is also valid for or or or .
Corollary 2.10.
Let
Then for and , we have
Proof.
Taking in (2.17), we get (2.22).
3. Mean Value Theorems
Lemma 3.1.
Let , where such that
Consider the functions and defined as
Then for are monotonically increasing functions.
Proof.
We have that
that is, for are monotonically increasing functions.
Theorem 3.2.
Let and be two positive tuples satisfy condition (1.6), all 's are not equal and let , where . Then there exists such that
Proof.
In Theorem 1.2, setting and , respectively, as defined in Lemma 3.1, we get the following inequalities:
Now by combining both inequalities, we get,
is nonzero, it is zero if equalities are given in conditions (1.6), that is, and .
Now by condition (3.1), there exist , such that
and (3.7) implies (3.4).
Theorem 3.3.
Let and be two positive tuples satisfy condition (1.6), all 's are not equal and let , where . Then there exists such that the following equality is true:
provided that the denominators are nonzero.
Proof.
Let a function be defined as
where and are defined as
Then, using Theorem 3.2 with , we have
Since
therefore, (3.11) gives
After putting values, we get (3.8).
Let be a strictly monotone continuous function then quasiarithmetic sum is defined as follows:
Theorem 3.4.
Let and be two positive tuples , all 's are not equal and let be strictly monotonic continuous functions, be positive strictly increasing continuous function, where and
Then there exists from such that
is valid, provided that all denominators are not zero.
Proof.
If we choose the functions and so that , and . Substituting these in (3.8),
Then by setting , we get (3.16).
Corollary 3.5.
Let and be two nonnegative tuples and let . Then
Proof.
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).
References
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.
Simić S: On logarithmic convexity for differences of power means. Journal of Inequalities and Applications 2007, 2007:8.
Anwar M, Pečarić JE: On logarithmic convexity for differences of power means. to appear in Mathematical Inequalities & Applications
Anwar M, Pečarić JE: New means of Cauchy's type. Journal of Inequalities and Applications 2008, 2008:10.
Acknowledgment
The authors are really very thankful to Mr. Martin J. Bohner for his useful suggestions.
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Pečarić, J., Rehman, A.U. On Logarithmic Convexity for Power Sums and Related Results. J Inequal Appl 2008, 389410 (2008). https://doi.org/10.1155/2008/389410
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DOI: https://doi.org/10.1155/2008/389410