- Research Article
- Open Access

# An Inequality for the Beta Function with Application to Pluripotential Theory

- Per Åhag
^{1}Email author and - Rafał Czyż
^{2}

**2009**:901397

https://doi.org/10.1155/2009/901397

© P. Åhag and R. Czyż. 2009

**Received:**4 June 2009**Accepted:**22 July 2009**Published:**19 August 2009

## Abstract

We prove in this paper an inequality for the beta function, and we give an application in pluripotential theory.

## Keywords

- Lebesgue Measure
- Unit Ball
- Algebraic Geometry
- Gamma Function
- Convex Cone

## 1. Introduction

Since Euler's days the research of these special functions and their generalizations have had great impact on, for example, analysis, mathematical physics, and statistics. In this paper we prove the following inequality for the beta function.

Inequality A.

If
*,* then we have equality in (1.4), and if
*,* then we have the opposite inequality for all
*,*

In Section 3 we will give an application of Inequality A within the pluripotential theory.

## 2. Proof of Inequality A

A crucial tool in Lemma 2.2 is the following theorem.

Theorem 2.1.

Proof.

This follows from [3, Theorem 8] (see also [4, 5]).

Lemma 2.2.

where is the digamma function. Then for all and all ( ). Furthermore, for all .

Proof.

From (2.6), (2.9), and the fact that , we conclude that for all and all ( ).

Proof.

Case 1 ( ).

which is precisely the desired equality.

Case 2 ( ).

Thus,

(a) has a minimum point in ;

(b) is decreasing on ;

(c) is increasing on ;

(d) .

Thus, for .

Case 3 ( ).

where is the digamma function. This proof is then completed by using Lemma 2.2.

## 3. The Application

*domain*is an open and connected set, and a bounded domain is

*hyperconvex*if there exists a plurisubharmonic function such that the closure of the set is compact in , for every ; that is, for every the level set is relatively compact in . The geometric condition that our underlying domain should be hyperconvex is to ensure that we have a satisfying quantity of plurisubharmonic functions. By we denote the family of all bounded plurisubharmonic functions defined on such that

If , then ([6, 7]). It should be noted that it follows from [6] that the complex Monge-Ampère operator is well defined on . For further information about pluripotential theory and the complex Monge-Ampère operator we refer to [8, 9].

The convex cone has applications in dynamical systems and algebraic geometry (see, e.g., [10, 11]). A fundamental tool in working with is the following energy estimate (the proof can be found in [12], see also [6, 13, 14]).

Theorem 3.1.

If for all functions in , then the methods in [15] would immediately imply that the vector space , with certain norm, is a Banach space. Furthermore, proofs in [15] (see also [6]) could be simplified, and some would even be superfluous. Therefore, it is important to know for which the constant is equal or strictly greater than one. With the help of Inequality A we settle this question. In Example 3.2, we show that there are functions such that, for all and all ( ), the constant , in (3.3), is strictly greater than .

Example 3.2.

This contradicts Inequality A. Thus, there are functions such that for all and all ( ).

## Declarations

### Acknowledgments

The authors would like to thank Leif Persson for fruitful discussions and encouragement. R. Czy was partially supported by ministerial Grant no. N N201 367933.

## Authors’ Affiliations

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## Copyright

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.