Some Equivalent Forms of the Arithematic-Geometric Mean Inequality in Probability: A Survey

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Research Article

Abstract

We link some equivalent forms of the arithmetic-geometric mean inequality in probability and mathematical statistics.

Keywords

Related Result Mathematical Statistic Equivalent Form Equivalent Relation Probabilistic Argument 

1. Introduction

The arithmetic-geometric mean inequality (in short, AG inequality) has been widely used in mathematics and in its applications. A large number of its equivalent forms have also been developed in several areas of mathematics. For probability and mathematical statistics, the equivalent forms of the AG inequality have not been linked together in a formal way. The purpose of this paper is to prove that the AG inequality is equivalent to some other renowned inequalities by using probabilistic arguments. Among such inequalities are those of Jensen, Hölder, Cauchy, Minkowski, and Lyapunov, to name just a few.

2. The Equivalent Forms

Let Open image in new window be a random variable, we define

where Open image in new window denotes the expected value of Open image in new window .

Throughout this paper, let Open image in new window be a positive integer and we consider only the random variables which have finite expected values.

In order to establish our main results, we need the following lemma which is due to Infantozzi [1, 2], Marshall and Olkin [3, Page 457], and Maligranda [4, 5]. For other related results, we refer to [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19].

Lemma 2.1.

The following inequalities are equivalent.

AG inequality: Open image in new window , where Open image in new window is a nonnegative random variable.
Artin's theorem. Let Open image in new window be an open convex subset of Open image in new window and Open image in new window satisfy

(a) Open image in new window is Borel-measurable in Open image in new window for each fixed Open image in new window

(b) Open image in new window is convex in Open image in new window for each fixed Open image in new window .

If Open image in new window is a measure on the Borel subsets of Open image in new window such that Open image in new window is Open image in new window integrable for each Open image in new window then Open image in new window is a convex function on Open image in new window .

Jensen's inequality. Let Open image in new window be a probability space and Open image in new window be a random variable taking values in the open convex set Open image in new window with finite expectation Open image in new window . If Open image in new window is convex, then Open image in new window .

Proof.

The proof of the equivalent relations of Open image in new window can be found in [1, 2, 4, 5].

The proof of the equivalent relations of Open image in new window , and Open image in new window can be found in [3].

Theorem 2.2.

The following inequalities are equivalent.

Cauchy-Bunyakovski and Schwarz's (CBS) inequality: Open image in new window if Open image in new window are random variables.
Minkowski's inequality: Open image in new window if Open image in new window are random variables, Open image in new window , and the opposite inequality holds if Open image in new window .
Triangle inequality: Open image in new window if Open image in new window are random variables, Open image in new window and the opposite inequality holds if Open image in new window .
if Open image in new window is a random variable, Open image in new window and Open image in new window are two continuous and strictly increasing functions such that Open image in new window is convex.

The above listed inequalities are also equivalent to the inequalities in Lemma 2.1.

Proof.

The sketch of the proof of this theorem is illustrated by the following maps of equivalent circles:

(1) Open image in new window ;

(2) Open image in new window ;

(3) Open image in new window ;

(4) Open image in new window ;

(5) Open image in new window ;

(6) Open image in new window ;

(7) Open image in new window ;

(8) Open image in new window ;

(9) Open image in new window .

Now, we are in a position to give the proof of this theorem as follows.

, see Casella and Berger [7, page 187].

Replacing Open image in new window by Open image in new window in the above inequality, we obtain Open image in new window .

Similarly, we can prove the case that Open image in new window and Open image in new window .

is proved similarly.
is similarly proved.

That is, Open image in new window holds.

is similarly proved.
which implies

This proves Open image in new window .

This proves Open image in new window holds.

is proved similarly.
which implies
which implies
which implies
  1. (d)

    It follows from (a), (b), and (c) that Open image in new window holds.

     

Thus, we complete the proof.

is similarly proved.

Replacing Open image in new window and Open image in new window by Open image in new window and Open image in new window in the above inequality, respectively, we obtain Open image in new window .

Casella and Berger [7, page 188].

This proves Open image in new window holds.

Therefore,
Letting Open image in new window in the both sides of the above inequality,

This shows Open image in new window (see [13]).

. First note that, as shown above, Open image in new window and Open image in new window are equivalent. It follows from Open image in new window that, for Open image in new window ,

These imply Open image in new window for the case Open image in new window .

Similarly, we can prove the case for Open image in new window or Open image in new window by using Open image in new window .

To complete our proof of equivalence of all inequalities in this theorem and in Lemma 2.1, it suffices to show further the following implications.

Remark 2.3.

That is,

is a decreasing function of Sclove et al. [18] proved this property by means of the convexity of Open image in new window , see [14]. Clearly, our method is simpler than theirs.

Remark 2.4.

Each Open image in new window (or Open image in new window ) is called Hölder's inequality, each Open image in new window (or Open image in new window ) is called CBS inequality, each Open image in new window is called Lyapunov's inequality, each Open image in new window is called Radon's inequality, each Open image in new window is related to Jensen's inequality.

Notes

Acknowledgments

The authors wish to thank three reviewers for their valuable suggestions that lead to substantial improvement of this paper. This work is dedicated to Professor Haruo Murakami on his 80th birthday.

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Copyright information

© Cheh-Chih Yeh et al. 2008

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Department of Information ManagementLunghwa University of Science and TechnologyTaoyuan CountyTaiwan
  2. 2.Department of BiostatisticsUniversity of KansasKansas CityUSA
  3. 3.Division of BiostatisticsUniversity of Texas-Health Science Center at HoustonHoustonUSA

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