On Logarithmic Convexity for Power Sums and Related Results

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 well-known Jensen Inequality [1, page 43] gives the following, for or :

(1.1)

and vice versa for .

Simić [2] has considered the difference of both sides of (1.1). He considers the function defined as

(1.2)

and has proved the following theorem.

Theorem 1.1.

For , then

(1.3)

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

(1.4)

Moreover, if is a positive -tuples such that , then for (see [1, page 165]), we have

(1.5)

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

(1.6)

If is an increasing function, then

(1.7)

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

(2.1)

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 ,

(2.2)

if and only if the relation

(2.3)

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

(2.4)

Theorem 2.4.

Let and be two positive -tuples and let

(2.5)

such that condition (1.6) is satisfied and all 's are not equal. Then is log-convex, also for where we have

(2.6)

Proof.

Since is a strictly increasing function for and all 's are not equal, therefore by Theorem 1.2 with , we have

(2.7)

that is, is a positive-valued function.

Let where and :

(2.8)

This implies that is monotonically increasing.

By Theorem 1.2, we have

(2.9)

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 ,

(2.10)

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 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.

Lemma 2.6.

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

(2.13)

Proof.

In [1, page 2], we have the following result for convex function , with :

(2.14)

Putting , we get

(2.15)

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

(2.16)

Remark 2.8.

Let us note that , and .

Theorem 2.9.

Let such that . Then we have

(2.17)

Proof.

Let

(2.18)

Now taking , where , and in Lemma 2.6, we have

(2.19)

Since by substituting and , where , in above inequality, we get

(2.20)

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

(2.21)

Then for and , we have

(2.22)

Proof.

Taking in (2.17), we get (2.22).

3. Mean Value Theorems

Lemma 3.1.

Let , where such that

(3.1)

Consider the functions and defined as

(3.2)

Then for are monotonically increasing functions.

Proof.

We have that

(3.3)

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

(3.4)

Proof.

In Theorem 1.2, setting and , respectively, as defined in Lemma 3.1, we get the following inequalities:

(3.5)

Now by combining both inequalities, we get,

(3.6)

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

(3.7)

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:

(3.8)

provided that the denominators are nonzero.

Proof.

Let a function be defined as

(3.9)

where and are defined as

(3.10)

Then, using Theorem 3.2 with , we have

(3.11)

Since

(3.12)

therefore, (3.11) gives

(3.13)

After putting values, we get (3.8).

Let be a strictly monotone continuous function then quasiarithmetic sum is defined as follows:

(3.14)

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

(3.15)

Then there exists from such that

(3.16)

is valid, provided that all denominators are not zero.

Proof.

If we choose the functions and so that , and . Substituting these in (3.8),

(3.17)

Then by setting , we get (3.16).

Corollary 3.5.

Let and be two nonnegative -tuples and let . Then

(3.18)

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).