- Research Article
- Open Access

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

- Cheh-Chih Yeh
^{1}Email author, - Hung-Wen Yeh
^{2}and - Wenyaw Chan
^{3}

**2008**:386715

https://doi.org/10.1155/2008/386715

© Cheh-Chih Yeh et al. 2008

**Received:**5 December 2007**Accepted:**24 June 2008**Published:**6 July 2008

## 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

where denotes the expected value of .

Throughout this paper, let 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 [6–19].

Lemma 2.1.

The following inequalities are equivalent.

(a) is Borel-measurable in for each fixed

(b) is convex in for each fixed .

If is a measure on the Borel subsets of such that is integrable for each then is a convex function on .

Proof.

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

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

Theorem 2.2.

The following inequalities are equivalent.

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) ;

(2) ;

(3) ;

(4) ;

(5) ;

(6) ;

(7) ;

(8) ;

(9) .

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

Replacing by in the above inequality, we obtain .

Similarly, we can prove the case that and .

That is, holds.

This proves .

This proves holds.

- (d)
It follows from (a), (b), and (c) that holds.

Thus, we complete the proof.

Replacing and by and in the above inequality, respectively, we obtain .

This proves holds.

This shows (see [13]).

These imply for the case .

Similarly, we can prove the case for or by using .

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.

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

Remark 2.4.

Each (or ) is called Hölder's inequality, each (or ) is called CBS inequality, each is called Lyapunov's inequality, each is called Radon's inequality, each is related to Jensen's inequality.

## Declarations

### 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.

## Authors’ Affiliations

## References

- Infantozzi CA:
**An introduction to relations among inequalities.***Proceedings of the American Mathematical Society Meeting 700, 1979, Cleveland, Ohio, USA*Google Scholar - Infantozzi CA:
**An introduction to relations among inequalities.***Notices of the American Mathematical Society*1972,**141:**A918-A820.Google Scholar - Marshall AW, Olkin I:
*Inequalities: Theory of Majorization and Its Applications, Mathematics in Science and Engineering*.*Volume 143*. Academic Press, New York, NY, USA; 1979:xx+569.Google Scholar - Maligranda L:
**Why Hölder's inequality should be called Rogers' inequality.***Mathematical Inequalities & Applications*1998,**1**(1):69–83.MATHMathSciNetView ArticleGoogle Scholar - Maligranda L:
**Equivalence of the Hölder-Rogers and Minkowski inequalities.***Mathematical Inequalities & Applications*2001,**4**(2):203–207.MATHMathSciNetView ArticleGoogle Scholar - Beckenbach EF, Bellman R:
*Inequality*. 4th edition. Springer, Berlin, Germany; 1984.Google Scholar - Casella G, Berger RL:
*Statistical Inference, Duxbury Advanced Series*. Thomson Learning, Duxbury, Mass, USA; 2002.Google Scholar - Callebaut DK:
**Generalization of the Cauchy-Schwarz inequality.***Journal of Mathematical Analysis and Applications*1965,**12**(3):491–494. 10.1016/0022-247X(65)90016-8MATHMathSciNetView ArticleGoogle Scholar - Chung KL:
*A Course in Probability Theory, Probability and Mathematical Statistics*.*Volume 2*. 2nd edition. Academic Press, New York, NY, USA; 1974:xii+365.Google Scholar - Gurland J:
**Inequalities of expectations of random variables derived by monotonicity or convexity.***The American Statistician*1968,**22**(2):26–27. 10.2307/2681985Google Scholar - Hardy GH, Littlewood JE, Pólya G:
*Inequalities*. 2nd edition. Cambridge University Press, Cambridge, UK; 1952:xii+324.MATHGoogle Scholar - Kendall M, Stuart A:
*The Advanced Theory of Statistics Vol 1*. 4th edition. Macmillan, New York, NY, USA; 1977.MATHGoogle Scholar - Li Y-C, Shaw S-Y:
**A proof of Hölder's inequality using the Cauchy-Schwarz inequality.***Journal of Inequalities in Pure and Applied Mathematics*2006,**7**(2):-3.MathSciNetGoogle Scholar - Loève M:
*Probability Theory*. 3rd edition. Springer, New York, NY, USA; 1998.Google Scholar - McLaughlin HW, Metcalf FT:
**Remark on a recent generalization of the Cauchy-Schwarz inequality.***Journal of Mathematical Analysis and Applications*1967,**18**(3):522–523. 10.1016/0022-247X(67)90043-1MATHMathSciNetView ArticleGoogle Scholar - Mitrinović DS:
*Analytic Inequalities*. Springer, New York, NY, USA; 1970:xii+400.MATHView ArticleGoogle Scholar - Mullen K:
**A note on the ratio of two independent random variables.***The American Statistician*1967,**21**(3):30–31.MathSciNetGoogle Scholar - Sclove SL, Simons G, Ryzin JV:
**Further remarks on the expectation of the reciprocal of a positive random variable.***The Americam Statistician*1967,**21**(4):33–34.Google Scholar - Zong XF:
**A further generalization of the Cauchy inequality.***Journal of Qufu Normal University*2000,**26**(4):32–34.MATHMathSciNetGoogle Scholar

## Copyright

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