ABSTRACT
The integral given in (3.1-7), with the exception of a different normalization [ q(O) = a ] has the same form as the integral given in (2.6-17). Since h is convex and Re 'YIn ~ 0' we deduce from part (ii) of Theorem 2.6h that q is convex and univalent. A simple calculation shows that q also satisfies the differential equation
Since q is the univalent solution of the differential equation (3.1-8)
associated with differential subordination (3.1-6), we can prove that it is the
best dominant by applying Theorem 2.3f. Without loss of generality, we can
assume that h and q are analytic and univalent on U, and q'(~) :t; 0 for I~ I = 1. If not, then we could replace h with hp(z) = h(pz), and q with qp(z) = q(pz). These new functions would then have the desired proper-
ties and we would prove the theorem using part (iii) of Theorem 2.3f.
