On Logarithmic Convexity for Power Sums and Related Results II

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 n-tuples. The well-known inequality for power sums of order and , for (see [1, page 164]), states that

(1.1)

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

(1.2)

In [2], we defined the following function:

(1.3)

We introduced the Cauchy means involving power sums. Namely, the following results were obtained in [2].

For where , we have

(1.4)

such that, and

(1.5)

We defined the following means.

Definition 1.1.

Let x and p be two nonnegative n-tuples such that . Then for ,

(1.6)

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 n-tuples such that condition (1.5) is valid. If is a convex function on , then

(1.7)

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

(2.1)

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

(2.2)

if and only if the relation

(2.3)

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

(2.4)

Lemma 2.4.

Let f be -convex function and if, , then the following inequality is valid:

(2.5)

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

(2.6)

such that condition (1.5) is satisfied and all 's are not equal. Then is -convex. Also for where , we have

(2.7)

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 n-tuples such that then for ,

(2.8)

Remark 2.7.

Let us note that , and .

Theorem 2.8.

Let

(2.9)

then for and , we have

(2.10)

Theorem 2.9.

Let , such that , . Then one has

(2.11)

Remark 2.10.

From (2.7), we have

(2.12)

Since is concave, therefore for , we have

(2.13)

This implies that (1.4), which we derived in [2], is better than (2.7).

Also note that

(2.14)

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

(3.1)

• (a) If

  

  (3.2)

then for every convex function such that for all ,

(3.3)

• (b) If and either there exists an such that

  

  (3.4)

or there exists an such that

(3.5)

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:

(3.6)

Theorem 3.3.

Let , defined as above, satisfy condition (3.2). Then is -convex. Also for , where , one has

(3.7)

Proof.

Let where and ,

(3.8)

This implies that is convex.

By Theorem 3.1, we have,

(3.9)

Now, by Lemma 2.2, we have is log-convex in Jensen sense.

Since , this implies that is continuous for all , therefore it is a log-convex [1, page 6].

Since is log-convex, that is, is convex, therefore by Lemma 2.3 for and taking , we have

(3.10)

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

(3.11)

Definition 3.5.

Let be fixed, be continuous and monotonic with , be a function of bounded variation. Then for , one defines

(3.12)

Remark 3.6.

Let us note that , and .

Theorem 3.7.

Let , such that , . Then

(3.13)

Proof.

Let

(3.14)

Now, taking , , , , where , and in Lemma 2.4, we have

(3.15)

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

(3.16)

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

(3.17)

Consider the functions , defined as

(3.18)

Then for are convex.

Proof.

We have that

(3.19)

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

(3.20)

Proof.

In Theorem 3.1, setting and , respectively, as defined in Lemma 3.8, we get the following inequalities:

(3.21)

(3.22)

Now, by combining both inequalities, we get

(3.23)

So by condition (3.17), there exists such that

(3.24)

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:

(3.25)

provided that denominators are nonzero.

Proof.

Let a function be defined as

(3.26)

where and are defined as

(3.27)

Then, using Theorem 3.9 with , we have

(3.28)

Since

(3.29)

therefore, (3.28) gives

(3.30)

After putting values, we get (3.25).

Let be a strictly monotone continuous function, we defined as follows (integral version of quasiarithmetic sum [2]):

(3.31)

Theorem 3.11.

Let be strictly monotonic continuous functions. Then there exists in the image of such that

(3.32)

is valid, provided that all denominators are nonzero.

Proof.

If we choose the functions and so that , , and . Substituting these in (3.25),

(3.33)

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

(3.34)

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.