ABSTRACT
Hence c > b and from Lemma 4.5e, we obtain F' (z) :1:- 0 in U . If we set p(z) = 1 + zF"(z)/ F'(z), then p is analytic in U and satisfies p(O) = 1. If we use this substitution in the ( Gaussian) hypergeometric differential equation
Since c > M(a, b) implies c-2-(a+ b)z :f:. 0 in U , we can rewrite (4.5-13) in the form
(4.5-14) J(z)[zp'(z) + p 2 (z)] + p(z) + [J(z)- K(z)]/2 = 0,
where
J(z) = 1 - z and K(z) = 2c - 1 - (1 - 2ab )z. c - 2 - (a + b )z c-2-(a+ b )z
A bilinear transformation w(z) = (1 + Az)/ (1 + Bz) , with - 1 $A$ 1 and - 1 $ B $ 1, has the property that Re w(z) > 0 for z E U. The conditions on a, b and c given in the hypothesis imply that J(z) and K(z)
are such bilinear transformations. Hence we have
(4.5-15) Re J(z) > 0 and Re K(z) > 0 for z E U.
