# 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

## 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.

which is precisely the desired equality.

Thus,

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

## Authors’ Affiliations

## References

- Davis PJ:
**Leonhard Euler's integral: a historical profile of the gamma function.***The American Mathematical Monthly*1959,**66:**849–869. 10.2307/2309786MathSciNetView ArticleMATHGoogle Scholar - Gautschi W:
**Leonhard Euler: his life, the man, and his works.***SIAM Review*2008,**50**(1):3–33. 10.1137/070702710MathSciNetView ArticleMATHGoogle Scholar - Alzer H:
**On some inequalities for the gamma and psi functions.***Mathematics of Computation*1997,**66**(217):373–389. 10.1090/S0025-5718-97-00807-7MathSciNetView ArticleMATHGoogle Scholar - Koumandos S:
**Remarks on some completely monotonic functions.***Journal of Mathematical Analysis and Applications*2006,**324**(2):1458–1461. 10.1016/j.jmaa.2005.12.017MathSciNetView ArticleMATHGoogle Scholar - Qi F, Cui R-Q, Chen C-P, Guo B-N:
**Some completely monotonic functions involving polygamma functions and an application.***Journal of Mathematical Analysis and Applications*2005,**310**(1):303–308. 10.1016/j.jmaa.2005.02.016MathSciNetView ArticleMATHGoogle Scholar - Cegrell U:
**Pluricomplex energy.***Acta Mathematica*1998,**180**(2):187–217. 10.1007/BF02392899MathSciNetView ArticleMATHGoogle Scholar - Cegrell U, Kołodziej S, Zeriahi A:
**Subextension of plurisubharmonic functions with weak singularities.***Mathematische Zeitschrift*2005,**250**(1):7–22. 10.1007/s00209-004-0714-4MathSciNetView ArticleMATHGoogle Scholar - Klimek M:
*Pluripotential Theory, London Mathematical Society Monographs*.*Volume 6*. The Clarendon Press, Oxford University Press, New York, NY, USA; 1991:xiv+266.MATHGoogle Scholar - Kołodziej S:
**The complex Monge-Ampère equation and pluripotential theory.***Memoirs of the American Mathematical Society*2005.,**178**(840):MATHGoogle Scholar - Åhag P, Cegrell U, Kołodziej S, Pham HH, Zeriahi A: Partial pluricomplex energy and integrability exponents of plurisubharmonic functions. Advances in Mathematics. In press Advances in Mathematics. In pressGoogle Scholar
- Diller J, Dujardin R, Guedj V: Dynamics of meromorphic maps with small topological degree II: energy and invariant measure. Commentrail Mathematici Helvetici. In press Commentrail Mathematici Helvetici. In pressGoogle Scholar
- Persson L:
**A Dirichlet principle for the complex Monge-Ampère operator.***Arkiv för Matematik*1999,**37**(2):345–356. 10.1007/BF02412219View ArticleMATHGoogle Scholar - Åhag P, Czyż R, Phąm HH:
**Concerning the energy class for .***Annales Polonici Mathematici*2007,**91**(2–3):119–130. 10.4064/ap91-2-2MathSciNetView ArticleGoogle Scholar - Cegrell U, Persson L:
**An energy estimate for the complex Monge-Ampère operator.***Annales Polonici Mathematici*1997,**67**(1):95–102.MathSciNetMATHGoogle Scholar - Åhag P, Czyż R: Modulability and duality of certain cones in pluripotential theory. Journal of Mathematical Analysis and Applications. In press Journal of Mathematical Analysis and Applications. In pressGoogle Scholar

## 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.