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Research Article An Inequality for the Beta Function with Application to Pluripotential Theory | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 901397 8 pages doi 10.1155 2009 901397 Research Article An Inequality for the Beta Function with Application to Pluripotential Theory Per Ahag1 and Rafaỉ Czyz2 1 Department of Natural Sciences Engineering and Mathematics Mid Sweden University 871 88 Harnosand Sweden 2 Institute of Mathematics Jagiellonian University Eojasiewicza 6 30-348 Krakow Poland Correspondence should be addressed to Per Ahag per.ahag@miun.se Received 4 June 2009 Accepted 22 July 2009 Recommended by Paolo Ricci We prove in this paper an inequality for the beta function and we give an application in pluripotential theory. Copyright 2009 P. Ahag and R. CzyZ. 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. 1. Introduction A correspondence that started in 1729 between Leonhard Euler and Christian Goldbach was the dawn of the gamma function that is given by r x e-tx-1dt 0 1.1 see e.g. 1 2 . One of the gamma function s relatives is the beta function which is defined by B a b f ta-1 1 - t b-1dt . 0 1.2 The connection between these two Eulerian integrals is B a b r a T b r a b 1.3 2 Journal of Inequalities and Applications 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 n e N and all p 0 p 0 p Ỷ1 there exists a number k 0 such that k n p np n p B p 1 kn B p 1 n . 1.4 If p 0 then we have equality in 1.4 and if p 1 then we have the opposite inequality for all n e N k 0. 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. .
