Duplication inequalities for the ratios of hypergeometric functions

The authors derive the analogs of the duplication formula and logarithmic i nequalities satisfied by mu(r), the modulus of the plane Grotzsch ring B-2\ [0,r], for its generalization mu(a)(r) = pi F(a, 1 - a; 1; 1 - r(2))/[2F(a, 1 - a; r(2)) sin(pi a)] for a is an element of (0, 1/2] and r is an elemen t of (0, 1). Sharp estimates are obtained for mu(a)(r), too.