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On Logarithmic Convexity for Power Sums and Related Results II
Journal of Inequalities and Applications volume 2008, Article number: 305623 (2008)
Abstract
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
Let x be positive ntuples. The wellknown inequality for power sums of order and , for (see [1, page 164]), states that
Moreover, if p is a positive ntuples such that then for (see [1, page 165]), we have
In [2], we defined the following function:
We introduced the Cauchy means involving power sums. Namely, the following results were obtained in [2].
For where , we have
such that, and
We defined the following means.
Definition 1.1.
Let x and p be two nonnegative ntuples such that . Then for ,
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].
Theorem 1.2.
Let and be two nonnegative ntuples such that condition (1.5) is valid. If is a convex function on , then
Remark 1.3.
In Theorem 1.2, if is strictly convex, then (1.7) is strict unless and .
2. Discrete Result
Lemma 2.1.
Let
where . Then is strictly convex for .
Here, we use the notation .
Proof.
Since for , therefore is strictly convex for .
Lemma 2.2 (see [3]).
A positive function is convex in Jensen 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 definition of convex function [1, page 2].
Lemma 2.3.
If is continuous and convex for all , , of an open interval for which , then
Lemma 2.4.
Let f be convex function and if, , then the following inequality is valid:
By using the above lemmas and Theorem 1.2, as in [2], we can prove the following results.
Theorem 2.5.
Let and be two positive tuples and let
such that condition (1.5) is satisfied and all 's are not equal. Then is convex. Also for where , we have
Moreover, we can use (2.7) to obtain new means of Cauchy type involving power sums.
Let us introduce the following means.
Definition 2.6.
Let x and p be two nonnegative ntuples such that then for ,
Remark 2.7.
Let us note that , and .
Theorem 2.8.
Let
then for and , we have
Theorem 2.9.
Let , such that , . Then one has
Remark 2.10.
From (2.7), we have
Since is concave, therefore for , we have
This implies that (1.4), which we derived in [2], is better than (2.7).
Also note that
Let us note that there are not integral analogs of results from [2]. 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].
Theorem 3.1.
Let be fixed, be continuous and monotonic with , be a function of bounded variation and

(a)
If
(3.2)
then for every convex function such that for all ,

(b)
If and either there exists an such that
(3.4)
or there exists an such that
then for every convex function such that for all , the reverse of the inequality in (3.3) holds.
To define the new means of Cauchy involving integrals, we define the following function.
Definition 3.2.
Let be fixed, be continuous and monotonic with , be a function of bounded variation. Choose such that function is positive valued, where is defined as follows:
Theorem 3.3.
Let , defined as above, satisfy condition (3.2). Then is convex. Also for , where , one has
Proof.
Let where and ,
This implies that is convex.
By Theorem 3.1, we have,
Now, by Lemma 2.2, we have is logconvex in Jensen sense.
Since , this implies that is continuous for all , therefore it is a logconvex [1, page 6].
Since is logconvex, that is, is convex, therefore by Lemma 2.3 for and taking , we have
which is equivalent to (3.7).
Theorem 3.4.
Let such that condition (3.4) or (3.5) is satisfied. Then is convex. Also for , where , one has
Definition 3.5.
Let be fixed, be continuous and monotonic with , be a function of bounded variation. Then for , one defines
Remark 3.6.
Let us note that , and .
Theorem 3.7.
Let , such that , . Then
Proof.
Let
Now, taking , , , , where , and in Lemma 2.4, we have
Since by substituting , and , where , in above inequality, we get
By raising power , we get an inequality (3.13) for .
From Remark 3.6 , we get (3.13) is also valid for or or or .
Lemma 3.8.
Let such that
Consider the functions , defined as
Then for are convex.
Proof.
We have that
that is, for are convex.
Theorem 3.9.
Let be fixed, be continuous and monotonic with , be a function of bounded variation, and such that condition (3.2) is satisfied. Then there exists such that
Proof.
In Theorem 3.1, setting and , respectively, as defined in Lemma 3.8, we get the following inequalities:
Now, by combining both inequalities, we get
So by condition (3.17), there exists such that
and (3.24) implies (3.20).
Moreover, (3.21) is valid if is bounded from above and again we have (3.20) is valid.
Of course (3.20) is obvious if is not bounded from above and below as well.
Theorem 3.10.
Let be fixed, be continuous and monotonic with , be a function of bounded variation, and such that condition (3.2) is satisfied. Then there exists such that the following equality is true:
provided that denominators are nonzero.
Proof.
Let a function be defined as
where and are defined as
Then, using Theorem 3.9 with , we have
Since
therefore, (3.28) gives
After putting values, we get (3.25).
Let be a strictly monotone continuous function, we defined as follows (integral version of quasiarithmetic sum [2]):
Theorem 3.11.
Let be strictly monotonic continuous functions. Then there exists in the image of such that
is valid, provided that all denominators are nonzero.
Proof.
If we choose the functions and so that , , and . Substituting these in (3.25),
Then by setting , we get (3.32).
Corollary 3.12.
Let be fixed, be continuous and monotonic with , be a function of bounded variation, and let . Then
Proof.
If and are pairwise distinct, then we put , and in (3.32) to get (3.34).
For other cases, we can consider limit as in Remark 3.6.
References
Pečarić J, 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.
Pečarić J, Rehman AU: On logarithmic convexity for power sums and related results. Journal of Inequalities and Applications 2008, 2008:9.
Simic S: On logarithmic convexity for differences of power means. Journal of Inequalities and Applications 2007, 2007:8.
Acknowledgments
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 11711708890888.
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Pečarić, J., ur Rehman, A. On Logarithmic Convexity for Power Sums and Related Results II. J Inequal Appl 2008, 305623 (2008). https://doi.org/10.1155/2008/305623
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DOI: https://doi.org/10.1155/2008/305623
Keywords
 Continuous Function
 Convex Function
 Positive Function
 Related Result
 Open Interval