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 .
- 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
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.