We give an extension of well-known Stolarsky means to the multivariable case in a simple and applicable way. Some basic inequalities concerning this matter are also established with applications in Analysis and Probability Theory.

1. Introduction

There is a huge amount of papers investigating properties of the so-called Stolarsky (or extended) two-parametric mean value, defined for positive values of , as

means can be continuously extended on the domain

by the following:

and in this form are introduced by Keneth Stolarsky in [1].

Most of the classical two-variable means are special cases of the class . For example, is the arithmetic mean, is the geometric mean, is the logarithmic mean, is the identric mean, and so forth. More generally, the th power mean is equal to .

Recently, several papers are produced trying to define an extension of the class to variables. Unfortunately, this is done in a highly artificial mode (cf. [24]), without a practical background. Here is an illustration of this point; recently Merikowski [4] has proposed the following generalization of the Stolarsky mean to several variables:

where is an -tuple of positive numbers and

The symbol stands for the Euclidean simplex which is defined by

In this paper, we give another attempt to generalize Stolarsky means to the multivariable case in a simple and applicable way. The proposed task can be accomplished by founding a “weighted” variant of the class , wherefrom the mentioned generalization follows naturally.

In the sequel, we will need notions of the weighted geometric mean and weighted th power mean , defined by


Note that for and .

1.1. Weighted Stolarsky Means

We introduce here a class of weighted two-parameters means which includes the Stolarsky class as a particular case. Namely, for , we define

Various properties concerning the means can be established; some of them are the following:

Note that

In the same manner, we get

The weighted means from the class can be extended continuously to the domain

This extension is given by

Note that those means are homogeneous of order that is, , symmetric in , but are not symmetric in unless .

1.2. Multivariable Case

A natural generalization of weighted Stolarsky means to the multivariable case gives

where , is an arbitrary positive weight sequence associated with and for .

We also write instead of .

The above formulae are obtained by an appropriate limit process, implying continuity.

For example, applying

we get

Remark 1.1. Analogously to the former considerations, one can define a class of Stolarsky means in variables as where .
Therefore, Details are left to the readers.

2. Results

The following basic assertion is of importance.

Proposition 2.1. The expressions are actual means, that is, for arbitrary weight sequence one has

Our main result is contained in the following.

Proposition 2.2. The means are monotone increasing in both variables and .

Passing to the continuous variable case, we get the following definition of the class .

Assuming that all integrals exist,

where is a positive integrable function and is a nonnegative function with .

From our former considerations, a very applicable assertion follows.

Proposition 2.3. is monotone increasing in either or .

3. Applications

3.1. Applications in Analysis

As an illustration of the above, we give the following proposition.

Proposition 3.1. The function defined by is monotone increasing for .
In particular, for one has where stands for the Gamma function, Zeta function, and Euler's constant, respectively.

3.2. Applications in Probability Theory

For a random variable and an arbitrary probability distribution with support on , it is well known that

Denoting the central moment of order by , we improve this inequality to the following propsositions.

Proposition 3.2. For an arbitrary probability law with support on , one has

Proposition 3.3. One also has that is monotone increasing in .

3.3. Shifted Stolarsky Means

Especially interesting is studying the shifted Stolarsky means , defined by

Their analytic continuation to the whole plane is given by

Main results concerning the means are contained in the following propositions.

Proposition 3.4. Means are monotone increasing in either or for each fixed

Proposition 3.5. Means are monotone increasing in either or for each .

A well known result of Qi ([5]) states that the means are logarithmically concave for each fixed and ; also, they are logarithmically convex for .

According to this, we propose the following proposition.

Open Question
Is there any compact interval such that the means are logarithmically convex (concave) for and each ?

A partial answer to this problem is given in what follows.

Proposition 3.6. On any interval which includes zero and ,
() are not logarithmically convex (concave);
() are logarithmically convex (concave) if and only if .

4. Proofs

For the proof of Proposition 2.1, we apply the following assertion on Jensen functionals from [6].

Theorem 4.1. Let be twice continuously differentiable functions. Assume that is strictly convex and is a continuous and strictly monotonic function on . Then the expression represents a mean value of the numbers , that is, if and only if the relation holds for each .

Recall that the Jensen functional is defined on an interval by

where , and is a positive weight sequence.

The famous Jensen's inequality asserts that

whenever is a (strictly) convex function on , with the equality case if and only if .

Proof of Proposition 2.1. Define the auxiliary function by Since we conclude that is a continuously twice differentiable convex function on .
Denoting , we realize that the condition (4.3) of Theorem 4.1 is fulfilled with . Hence, applying Theorem 4.1, we obtain that represents a mean value, which is equivalent to the assertion of Proposition 2.1.

Proof of Proposition 2.2. We prove first a global theorem concerning log-convexity of the Jensen's functional with a parameter, which can be very usable (cf. [7]).Theorem 4.2. Let be a twice continuously differentiable function in with a parameter . If is log-convex in for , then the Jensen functional is log-convex in for , where is any positive weight sequence.At the beginning, we need some preliminary lemmas. Lemma 4.3. A positive function is log-convex on if and only if the relation holds for each real and . This assertion is nothing more than the discriminant test for the nonnegativity of second-order polynomials. Other well known assertions are the following (cf [8, pages 74, 97-98]) lemmas.Lemma 4.4 (Jensen's inequality). If is twice continuously differentiable and on , then is convex on and the inequality holds for each and any positive weight sequence . Lemma 4.5. For a convex , the expression is increasing in both variables.Proof of Theorem 4.2. Consider the function defined as where are real parameters independent of the variable .
Since and by assuming is log-convex in , it follows from Lemma 4.3 that .
Therefore, by Lemma 4.4, we get which is equivalent to
According to Lemma 4.3 again, this is possible only if is log-convex and the proof is done.

Now, the proof of Proposition 2.2 easily follows.
From the above, we see that is twice continuously differentiable and that is a log-convex function for each real .
Applying Theorem 4.2, we conclude that the form is log-convex in .
By Lemma 4.5, with , we find out that is monotone increasing either in or . Therefore, by changing variable , we finally obtain the proof of Proposition 2.2.

Proof of Proposition 2.3. The assertion of Proposition 2.3 follows from Proposition 2.2 by the standard argument (cf. [8, pages 131–134]). Details are left to the reader.

Proof of Proposition 3.1. The proof follows putting and applying Proposition 2.2. with . Corresponding integrals are with

Proof of Proposition 3.2. By Proposition 2.3, we get that is, Using the identity , we obtain the proof of Proposition 3.2.

Proof of Proposition 3.3. This assertion is straightforward consequence of the fact that is monotone increasing in .

Proof of Proposition 3.4. Direct consequence of Proposition 2.2.

Proof of Proposition 3.5. This is left as an easy exercise to the readers.

Proof of Proposition 3.6. We prove only part (ii). The proof of (i) goes along the same lines.
Suppose that and that are log-convex (concave) for and any fixed . Then there should be an such that is of constant sign for each .
Substituting , after some calculations, we get that the above is equivalent to the assertion that is of constant sign, where Developing in power series in , we get
Therefore, can be of constant sign for each only if .
Suppose now that is of the form or . Then there should be an such that is of constant sign for each .
Proceeding as before, this is equivalent to the assertion that is of constant sign with However,
Hence, we conclude that can be of constant sign for sufficiently small only if . Combining this with Feng Qi theorem, the assertion from Proposition 3.6 follows.