On logarithmic concavity of series in gamma ratios

We find two-sided bounds and prove non-negativeness Taylor coefficients for the Turán determinants power series with coefficients involving the ratio of gamma-functions. We consider such series as functions of simultaneous shifts of the arguments of the gamma-functions located in the numerator and the denominator. These results are then applied to derive new inequalities for the Gauss hypergeometric function, the incomplete normalized beta-function and the generalized hypergeometric series. This communication continues the research of various authors who investigated logarithmic convexity and concavity of hypergeometric functions in parameters.