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  • Research Article
  • Open Access

On Logarithmic Convexity for Power Sums and Related Results

Journal of Inequalities and Applications20082008:389410

  • Received: 28 March 2008
  • Accepted: 29 June 2008
  • Published:


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

1. Introduction and Preliminaries

Let , denote two sequences of positive real numbers with . The well-known 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 well-known 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.

  1. (i)
    there exists an such that
where and , then (1.7) holds.
  1. (ii)
    If there exists an such that

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.


Then is a strictly increasing function for .


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 log-convex, also for where we have


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 positive-valued 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 log-convex in Jensen sense.

Since it follows that is continuous, therefore it is a log-convex function [1, page 6].

Since is log-convex, 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

  1. (i)
    if such that condition (1.8) is satisfied, then is log-convex. Also for , we have
  1. (ii)
    moreover if and (1.9) is satisfied, then we have that is log-convex and

We will also use the following lemma.

Lemma 2.6.

Let f be a log-convex function and assume that if . Then the following inequality is valid:


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


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.

Then for and , we have


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.


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


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.


Let a function be defined as
where and are defined as
Then, using Theorem 3.2 with , we have
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.


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


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.

Authors’ Affiliations

Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia
Abdus Salam School of Mathematical Sciences, GC University, Lahore, 54660, Pakistan


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  3. Anwar M, Pečarić JE: On logarithmic convexity for differences of power means. to appear in Mathematical Inequalities & ApplicationsGoogle Scholar
  4. Anwar M, Pečarić JE: New means of Cauchy's type. Journal of Inequalities and Applications 2008, 2008:-10.Google Scholar


© J. Pečarić and A. U. Rehman. 2008

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