- Research Article
- Open Access
An Inequality for the Beta Function with Application to Pluripotential Theory
© P. Åhag and R. Czyż. 2009
Received: 4 June 2009
Accepted: 22 July 2009
Published: 19 August 2009
We prove in this paper an inequality for the beta function, and we give an application in pluripotential theory.
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.
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.
which is precisely the desired equality.
3. The Application
If , then ([6, 7]). It should be noted that it follows from  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 , see also [6, 13, 14]).
If for all functions in , then the methods in  would immediately imply that the vector space , with certain norm, is a Banach space. Furthermore, proofs in  (see also ) 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 .
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