Turán type inequalities for generalized complete elliptic integrals

In this paper our aim is to establish some Turán type inequalities for Gaussian hypergeometric functions and for generalized complete elliptic integrals. These results complete the earlier result of P. Turán proved for Legendre polynomials. Moreover we show that there is a close connection between a Turán type inequality and a sharp lower bound for the generalized complete elliptic integral of the first kind. At the end of this paper we prove a recent conjecture of T. Sugawa and M. Vuorinen related to estimates of the hyperbolic distance of the twice punctured plane. Dedicated to my son Koppány.