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.