Normalized incomplete beta function: log-concavity in parameters and other properties

TL;DR

This paper provides a standard-analytic proof of the log-concavity of the normalized incomplete beta function in its parameters, investigates Turán determinants, and introduces new identities and bounds related to these functions.

Contribution

It offers a direct analytic proof of log-concavity, explores Turán determinants, and presents new combinatorial identities and bounds for the normalized incomplete beta function.

Findings

Established log-concavity using standard analysis methods.

Demonstrated sign consistency of Turán determinants under certain conditions.

Derived new combinatorial identities and bounds for the functions.

Abstract

The normalized incomplete beta function can be defined either as cumulative distribution function of beta density or as the Gauss hypergeometric function with one of the upper parameters equal to unity. Logarithmic concavity/convexity of this function in parameters was established by Finner and Roters in 1997. Their proof is indirect and rather difficult; it is based on generalized reproductive property of certain more general distributions. These authors remark that these results "seems to be very hard to obtain by usual analytic methods". In the first part of this paper we provide such proof based on standard tools of analysis. In the second part we go one step further and investigate the sign of generalized Tur\'{a}n determinants formed by shifts of the normalized incomplete beta function. Under some additional restrictions we demonstrate that these coefficients are of the same sign. We further conjecture that such restrictions can be removed without altering the results. Our method of proof also leads to various companion results which may be of independent interest. In particular, we establish linearization formulas and two-sided bounds for the above mentioned Tur\'{a}n determinants. Further, we find two combinatorial style identities for finite sums which we believe to be new.