Normalized Incomplete Beta Function: Log-Concavity in Parameters and Other Properties

Abstract

The logarithmic concavity/convexity in parameters of the normalized incomplete beta function has been established by Finner and Roters in 1997 as a corollary of a rather difficult result, based on generalized reproductive property of certain distributions. In the first part of this paper, a direct analytic proof of the logarithmic concavity/convexity mentioned above is presented. These results are strengthened in the second part, where it is proved that the power series coefficients of the generalized Turán determinants formed by the parameter shifts of the normalized incomplete beta function have constant sign under some additional restrictions. The method of proof suggested also leads to various other new facts, which may be of independent interest. In particular, linearization formulas and two-sided bounds for the above-mentioned Turán determinants, and also two identities of combinatorial type, which we believe to be new, are established.