Some Equivalent Forms of the Arithematic-Geometric Mean Inequality in Probability: A Survey
© Cheh-Chih Yeh et al. 2008
Received: 5 December 2007
Accepted: 24 June 2008
Published: 6 July 2008
We link some equivalent forms of the arithmetic-geometric mean inequality in probability and mathematical statistics.
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].
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 .
The proof of the equivalent relations of , and can be found in .
The following inequalities are equivalent.
The above listed inequalities are also equivalent to the inequalities in Lemma 2.1.
The sketch of the proof of this theorem is illustrated by the following maps of equivalent circles:
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.
- (b)Taking and in ,(2.12)
- (c)Taking and in ,(2.14)
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 ).
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.
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.
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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