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

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 901397 (2009)

## Abstract

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

## 1. Introduction

A correspondence that started in 1729 between LeonhardEuler and Christian Goldbach was the dawn of the gamma function that is given by

(see, e.g., [1, 2]). One of the gamma function's relatives is the beta function, which is defined by

The connection between these two Eulerian integrals is

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.

For all and all (*,*) there exists a number such that

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.

Let be the digamma function. Then for it holds that

Proof.

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

Lemma 2.2.

Let be a function defined by

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

Proof.

Since we have that , and

From the construction of we also have that . By using (2.3) we get that

From Theorem 2.1 it follows that

Thus,

Furthermore,

and since (Theorem 2.1), we get that

which means that

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

Proof.

Case 1 ().

The definition

yields that . Thus,

which is precisely the desired equality.

Case 2 ().

We will now prove that for all it holds that

Inequality (2.12) is equivalent to

Hence, to complete this case we need to prove that for all we have that

Let be defined by

To obtain (2.14) it is sufficient to prove that . The definition of yields that

Thus,

(a) has a minimum point in ;

(b) is decreasing on ;

(c) is increasing on ;

(d).

Thus, for .

Case 3 ().

Fix . Let be the function defined by

This construction implies that is continuously differentiable, and . To prove this case it is enough to show that . By rewriting with (1.3) the function can be written as

and therefore we get that

Thus

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

## 3. The Application

We start this section by recalling some definitions and needed facts. A *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

where is the complex Monge-Ampère operator. Next let , , denote the family of plurisubharmonic functions defined on such that there exists a decreasing sequence , , that converges pointwise to on , as tends to , and

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.

Let , and . Then there exists a constant , depending only on and , such that for any it holds that

Moreover,

and . If , then one follows [12] and interprets (3.3) as

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.

Let be the unit ball, and for set

Hence,

where is the Lebesgue measure on . For we then have that

where is the Lebesgue measure on . If , then

If we assume that in Theorem 3.1, then it holds that

Hence,

In particular, if , then we get that

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

## References

- [1]
Davis PJ:

**Leonhard Euler's integral: a historical profile of the gamma function.***The American Mathematical Monthly*1959,**66:**849–869. 10.2307/2309786 - [2]
Gautschi W:

**Leonhard Euler: his life, the man, and his works.***SIAM Review*2008,**50**(1):3–33. 10.1137/070702710 - [3]
Alzer H:

**On some inequalities for the gamma and psi functions.***Mathematics of Computation*1997,**66**(217):373–389. 10.1090/S0025-5718-97-00807-7 - [4]
Koumandos S:

**Remarks on some completely monotonic functions.***Journal of Mathematical Analysis and Applications*2006,**324**(2):1458–1461. 10.1016/j.jmaa.2005.12.017 - [5]
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.2005.02.016 - [6]
Cegrell U:

**Pluricomplex energy.***Acta Mathematica*1998,**180**(2):187–217. 10.1007/BF02392899 - [7]
Cegrell U, Kołodziej S, Zeriahi A:

**Subextension of plurisubharmonic functions with weak singularities.***Mathematische Zeitschrift*2005,**250**(1):7–22. 10.1007/s00209-004-0714-4 - [8]
Klimek M:

*Pluripotential Theory, London Mathematical Society Monographs*.*Volume 6*. The Clarendon Press, Oxford University Press, New York, NY, USA; 1991:xiv+266. - [9]
Kołodziej S:

**The complex Monge-Ampère equation and pluripotential theory.***Memoirs of the American Mathematical Society*2005.,**178**(840): - [10]
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 press - [11]
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 press - [12]
Persson L:

**A Dirichlet principle for the complex Monge-Ampère operator.***Arkiv för Matematik*1999,**37**(2):345–356. 10.1007/BF02412219 - [13]
Åhag P, Czyż R, Phąm HH:

**Concerning the energy class for .***Annales Polonici Mathematici*2007,**91**(2–3):119–130. 10.4064/ap91-2-2 - [14]
Cegrell U, Persson L:

**An energy estimate for the complex Monge-Ampère operator.***Annales Polonici Mathematici*1997,**67**(1):95–102. - [15]
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 press

## 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.

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Åhag, P., Czyż, R. An Inequality for the Beta Function with Application to Pluripotential Theory.
*J Inequal Appl* **2009, **901397 (2009). https://doi.org/10.1155/2009/901397

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### Keywords

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