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.

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

Most of the classical two-variable means are special cases of the class

Recently, several papers are produced trying to define an extension of the class

The symbol

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

In the sequel, we will need notions of the weighted geometric mean

We introduce here a class

In the same manner, we get

The weighted means from the class

Note that those means are homogeneous of order

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

We also write

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

For example, applying

Analogously to the former considerations, one can define a class of Stolarsky means in

Therefore,

The following basic assertion is of importance.

The expressions

Our main result is contained in the following.

The means

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

Assuming that all integrals exist,

From our former considerations, a very applicable assertion follows.

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

The function

In particular, for

For a random variable

Denoting the central moment of order

For an arbitrary probability law with support on

One also has that

Especially interesting is studying the

Main results concerning the means

Means

Means

A well known result of Qi ([

According to this, we propose the following proposition.

Is there any compact interval

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

On any interval

(

(

For the proof of Proposition

Let

Recall that the Jensen functional

The famous Jensen's inequality asserts that

Define the auxiliary function

Denoting

We prove first a global theorem concerning log-convexity of the Jensen's functional with a parameter, which can be very usable (cf. [

Let

A positive function

If

For a convex

Consider the function

Since

Therefore, by Lemma

According to Lemma

Now, the proof of Proposition

From the above, we see that

Applying Theorem

By Lemma

The assertion of Proposition

The proof follows putting

By Proposition

This assertion is straightforward consequence of the fact that

Direct consequence of Proposition

This is left as an easy exercise to the readers.

We prove only part (ii). The proof of (i) goes along the same lines.

Suppose that

Substituting

Therefore,

Suppose now that

Proceeding as before, this is equivalent to the assertion that

Hence, we conclude that