1. Introduction

Let be a fixed natural number, and an interval of real numbers. For every , the arithmetic mean associated to is defined as:

Let be an interval. If is a convex (concave) function, then the well known Jensen inequality (see [2]-[4], [10], [11] says that:

which can be written, using the above notation, as:

(1.1)

The inequality (1.1) represents a relation between the image of , through , and the arithmetic mean of the images of the numbers

We can imagine the numbers as being some data received by a statistician. For large values of , it is hard for the statistician to look at each number, in these data. So, the statistician decides to make a skeleton of the data, composed of the minimum and maximum values of these data, a center of the data, and a number measuring the spread of the data. While and are easy to define as:

(1.2)

and

(1.3)

there are many different ways to define the center . One very popular way is to a apply first a strictly monotone function to the data , obtaining the new data . Then we compute the average value , defined as:

(1.4)

(1.5)

Finally, we apply the inverse function to , defining the center of the data to be:

(1.6)

A typical example of such functions, used in computing the center of data consisting of positive numbers, is given by the functions: , for . Following the above procedure, these functions give rise to the H¨older means of the positive numbers

Because different functions and produce different centers and of the data, it is important to find inequalities relating and in order to understand the inequalities between and .

2. Main Result

We present now the result leading to the main result of this note.

Proposition 2.1 If

1. is a convex (concave) function;

2. , for all , where

then the following inequality:

(2.7)

holds.

Proof. Let be fixed. Since can be written as the following convex combination of and

If we denote and , then since , and we conclude from the definition of convexity of , that:

(2.8)

Summing up in the last inequality from we obtain:

After multiplying both sides of the last relation by we obtain:

which is equivalent to:

If is concave all the inequalities from this proof are reversed.

Corollary 2.2. If we impose the additional condition that to the assumptions from Proposition 2.1, then:

(2.9)

Proof. Since , we have . Dividing both sides of the inequality (2.7) by the strictly positive number , we obtain (2.9).

Theorem 2.3 Let be real numbers, and . Let and , be two strictly increasing and bijective functions, such that: is convex on . Then:

(2.10)

Proof. We apply Proposition 2.1 to the numbers: and the convex function obtaining inequality (2.10).

Application 2.4 Let It is clear that is convex and strictly increasing on . Thus for all , such that , we conclude from (2.9), that:

(2.11)

If , with at least one of these inequalities being strict, then setting: , we can rewrite (2.11) as:

(2.12)

Corollary 2.5 If , for all then:

where is the geometric mean of the numbers

Proof. Since the function is concave, it follows from the inequality (2.7) that:

If then we can take and obtain:

(2.13)

This inequality holds even when

Application 2.6 Let , for all Then we can take and We have:

It follows now from (2.13) that:

This is equivalent to:

Corollary 2.7 If for all and then:

(2.14)

where

is the harmonic mean of the numbers

Proof. The function is convex. Applying the inequality (2.7) to this function, we get:

If then by taking and in (2.14), we obtain:

Alternative characterizations for means, obtained by different methods, can be found in [6]-[9].

Observation 2.8 The means and make sense for negative numbers, too. If , then the inequality (2.14) is reversed, which can be proved very simply by multiplying both sides of this inequality by -1.

Application 2.9 Let , for all . Then we can take , and we have . It follows now from (2.14) that:

This is equivalent to

The last inequality means:

Corollary 2.10 Let and . Then the following inequality holds:

(2.15)

where:

is the power mean of order (called also the H¨older mean of order ) of the numbers In particular if and then:

(2.16)

Proof. Everything follow by applying the inequality (2.7) to the convex and increasing function , .

Application 2.11 For all and all positive numbers we have:

where denotes the mean of for all

Proof. We simply apply Theorem 2.3 to the numbers and the function and . We have , which is a convex function since , and thus the result of our application follows easily.

Observation 2.12 All the inequalities from this paper can be reformulated using weighted means, too.

ACKNOWLEDGEMENTS

• The author would like to thank the referees for their kind comments and suggestions that greatly helped him in improving the quality of this paper.

• The author would also like to thank Professor Aurel I. Stan, from The Ohio State University, for helping him in preparing this manuscript.