Abstract and Applied Analysis

Volume 2012 (2012), Article ID 203145, 25 pages

http://dx.doi.org/10.1155/2012/203145

## The Inequalities for Quasiarithmetic Means

^{1}Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lučića 5, 10000 Zagreb, Croatia^{2}Mechanical Engineering Faculty, University of Osijek, Trg Ivane Brlić Mažuranić 2, 35000 Slavonski Brod, Croatia^{3}Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovića 30, 10000 Zagreb, Croatia

Received 15 March 2012; Accepted 9 June 2012

Academic Editor: Sergey V. Zelik

Copyright © 2012 Jadranka Mićić et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Overview and refinements of the results are given for discrete, integral, functional and operator variants of inequalities for quasiarithmetic means. The general results are applied to further refinements of the power means. Jensen's inequalities have been systematically presented, from the older variants, to the most recent ones for the operators without operator convexity.

#### 1. Introduction

Quasiarithmetic means are very important because they are general and unavoidable in applications. This paper begins with the quasiarithmetic means of points, continues with the quasiarithmetic means of measurable function, through the quasiarithmetic means of functions with respect to linear functionals, and ends with the quasiarithmetic means of operators with respect to linear mappings. Conclusion of the paper is dedicated to the applications of operator quasiarithmetic means on power means with strictly positive operators. At this point, it should be emphasized that in all four of the next sections the basic and initial inequality was precisely the Jensen inequality (see Figure 1).

The applications of convexity often used strictly monotone continuous functions

Through this paper, we suppose that

#### 2. Results for Basic Case

For

Below is a discrete basic form of Jensen’s inequality for a convex function with respect to convex combinations points in interval.

Theorem A. *Let *

*A function* is convex if and only if the following inequality

*holds for all above* -tuples and .

*Consequently, if* , not necessarily equals , then is convex if and only if

*A function* is concave if and only if the reverse inequality is valid in (2.1) and (2.2).

A function

Let

Basic quasiarithmetic means have the property

Suppose that all coefficients

Corollary 2.1. *Let *

*A function* is either -convex and increasing or -concave and decreasing if and only if following the inequality:

*holds for all* -tuples and as in (2.3).

*A function* is either -concave and increasing or -convex and decreasing if and only if the reverse inequality is valid in (2.6).

A function

Suppose that all

Recall that a function

Let

Lemma 2.2. *Let *

*If* and , then

*provided that denominators are the same sign. The inequality in (2.11) is strict if* and .

*If either* and or and , then the reverse inequality is valid in (2.11).

Proposition 2.3. *Let *

*If* is -convex (resp. -concave), then is -convex (resp. -concave).

*Proof. *Suppose that

If

According to Proposition 2.3, we can express refinements of the basic quasiarithmetic means.

Theorem 2.4. *Let *

*If either* is -convex with both and increasing or -concave with both and decreasing, then the following inequality:

*holds for all* -tuples and as in (2.3).

*If either* is -concave with both and increasing or -convex with both and decreasing, then the reverse inequality is valid in (2.14).

*Proof. *If

In other words, the above theorem says that a function

We emphasize that the inequality in (2.14) is strict for

Let us take strictly monotone decreasing functions

Let us take strictly monotone increasing functions

Connecting two above inequalities results in

The inequality in (2.20) is strict for

The weighted harmonic-geometric-arithmetic inequality is only the special case of a whole collection of inequalities which can be derived by applying of Corollary 2.1 on power means. As a special case of the basic quasiarithmetic mean in (2.3) with

Very useful consequence of Corollary 2.1 is a well-known property of monotonicity of basic power means.

Corollary 2.5. *If *

*holds for all* -tuples and as in (2.3) with .

The inequality in (2.22) is strict for

Let functions

*Case *

Functions

*Case *

Functions

*Case *

Functions

*Case *

Functions

Given traditional signs of power means, we will mark

Corollary 2.6. *Let *

*If* or or or , then the inequality

*holds for all* -tuples and as in (2.3) with .

*If* , then we can take the series of inequalities

The inequalities in (2.24)-(2.25) are strict for

The inequality in (2.20) is a special case of the collection of inequalities in (2.24).

#### 3. Applications on Integral Case

In this section,

For

Here is an integral form of Jensen’s inequality for a convex function with respect to measurable functions with weighted functions on the probability measure space.

Theorem B. *Let *

*If a function* is convex, then the inequality

*holds for all above* , and .

*Consequently, if* , not necessarily equals , then

*If a function* is concave, then the reverse inequality is valid in (3.1) and (3.2).

The assumption

*Remark 3.1. *The reverse of Theorem B is valid if for any

Theorem B can be generalized to

Theorem 3.2. *Let *

*A function* is convex if and only if the inequality

*holds for all above* -tuples , and .

*Consequently, if* , not necessarily equals , then is convex if and only if

*A function* is concave if and only if the reverse inequality is valid in (3.6) and (3.7).

In the proof of sufficiency theorem, we simply take

A function

Let

Bearing in mind Theorem 3.2, the following corollary is valid.

Corollary 3.3. *Let *

*A function* is either -convex and increasing or -concave and decreasing if and only if the inequality

*holds for all* -tuples , , and as in (3.8).

*A function* is either -concave and increasing or -convex and decreasing if and only if the reverse inequality is valid in (3.10).

Combining basic and integral case by Corollaries 2.1 and 3.3, we get the following.

Proposition 3.4. *Let *

*holds for all* -tuples and as in (2.3) if and only if the inequality

*holds for all* -tuples , , and as in (3.8).

The one direction of Proposition 3.4 is proved in [2, Theorem 1]. It is proved that the inequality for basic case implies the inequality for integral case with one function

The following integral analogy of Theorem 2.4.

Theorem 3.5. *Let *

*If either* is -convex with both and increasing or -concave with both and decreasing, then the inequality

*holds for all* -tuples , , and as in (3.8).

*If either* is -concave with both and increasing or -convex with both and decreasing, then the reverse inequality is valid in (3.13).

The inequality in (3.13) is strict for

An integral version of refinements of the harmonic-geometric-arithmetic inequality is also valid. So, the inequality

As a special case of the integral quasiarithmetic mean in (3.8) with

We quote the integral analogy of Corollary 2.6. The following is the property of monotonicity, with refinements, of integral power means.

Corollary 3.6. *Let *

*If* or or or , then the inequality

*holds for all* -tuples , , and as in (3.8) with .

*If* , then we can take the series of inequalities

The inequalities in (3.16)-(3.17) are strict for

All the observed integral cases are reduced to the corresponding basic cases when we take constants

#### 4. Applications on Functional Case

Let

In this section, it is assumed that every weighted function

Bellow is a functional form of Jensen’s inequality for a convex function with respect to real-valued functions with weighted functions on the vector space of real-valued functions.

Theorem C. *Let *

*If a function* is convex, then the inequality

*holds for all above* , , and .

*Consequently, if* , not necessarily equals , then

*If a function* is concave, then the reverse inequality is valid in (4.1) and (4.2).

The inequality in (4.1) with

The interval

*Example 4.1. *Let

*Remark 4.2. *Suppose that

Theorem C can be generalized to

Theorem 4.3. *Let *

*If a function* is convex, then the inequality

*holds for all above* -tuples , , and .

*Consequently, if* , not necessarily equals , then

*If a function* is concave, then the reverse inequality is valid in (4.6) and (4.7).

*Proof. *Let us prove the inequality in (4.6). If

If

*Remark 4.4. *Suppose that

Let

Corollary 4.5. *Let *

*If a function* is either -convex and increasing or -concave and decreasing, then the inequality

*holds for all* -tuples , , and as in (4.9).

*If a function* is either -concave and increasing or -convex and decreasing, then the reverse inequality is valid in (4.12).

*Proof. *Suppose that

According to Remark 4.2, the reverse of Corollary 4.5 is valid if

Proposition 4.6. *Let *

*holds for all* and as in (2.3) if and only if the inequality

*holds for all* -tuples , , and as in (4.9) with and unital functionals .

Next in line is a functional analogy of refinements.

Theorem 4.7. *Let *

*If either* is -convex with both and increasing or -concave with both and decreasing, then the inequality

*holds for all* -tuples , , and as in (4.9) with and unital functionals .

*If either* is -concave with both and increasing or -convex with both and decreasing, then the reverse inequality is valid in (4.17).

The inequality in (4.17) is strict for

A functional version of refinements of the harmonic-geometric-arithmetic inequality is also valid. So, the inequality

As a special case of the functional quasiarithmetic mean in (4.9) with

The following is the property of monotonicity, with refinements, of functional power means.

Corollary 4.8. *Let *

*If* or or or , then the inequality

*holds for all* -tuples , , and as in (4.9) with , and unital functionals .

*If* , then we can take the series of inequalities

The inequalities in (4.20)-(4.21) are strict for

All the observed functional cases are reduced to the corresponding integral cases when we take

#### 5. Results for Operator Case

We recall some notations and definitions. Let

Let

A continuous function

For

In this section, it is assumed that every weighted operator

From the second half of the last century, Jensen’s inequality was formulated for operator convex functions, self-adjoint operators, and positive linear mappings (see [5–8]). Very recently, Jensen’s inequality for operators without operator convexity is formulated in [3], and generalized in [4].

The following theorem essentially coincides with the main theorem in [3]. The only difference is that now we add the weighted operators. We also give a short proof of the theorem that relies on the geometric property of convexity and affinity of the chord line or support line. So, we start with an operator form of Jensen’s inequality for a convex function with respect to self-adjoint operators with weighted operators on the Hilbert space, and positive linear mappings.

Theorem 5.1. *Let *

*If a function* is convex, then the inequality

*holds for all above* -tuples , , and provided spectral conditions

*Consequently, if* is strictly positive, not necessarily equals , then

*If a function* is concave, then the reverse inequality is valid in (5.4) and (5.6).

*Proof. *If

If

If

*Remark 5.2. *The reverse of Theorem 5.1 is trivially valid if all

Let

Corollary 5.3. *Let *

*Let* , and be as in (5.8). Let and be bounds of operators and , respectively.

*If a function* is either -convex with operator increasing or -concave with operator decreasing , then the inequality

*holds for all above* -tuples , , and provided spectral conditions

*If a function* is either -concave with operator increasing or -convex with operator decreasing , then the reverse inequality is valid in (5.11).

The following is operator analogy of Theorem 2.4.

Theorem 5.4. *Let *

*If either* is -convex with operator increasing , , and or -concave with operator decreasing , , and , then the inequality

*holds for all* -tuples , , and as in (5.8) that provided the following spectral conditions:

*If either* is -concave with operator increasing , , and or -convex with operator decreasing , , and , then the reverse inequality is valid in (5.13).

*Proof. *Let us prove the middle part of the inequality in (5.13), one that refers to

The inequality in (5.13) is strict for

We are interested in sufficient conditions under which the functions

Lemma 5.5. *Let *

*where* are nonnegative continuous functions such that for every .

*Proof. *Take any