  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.
