ABSTRACT
Next suppose that f -/. g. Then there is a k e {1, 2, · · · , n} such that A -/. Fk. We can assume that the univalency of the function Fk on U can be extended to U, since if this is not the case we can continue the proof by a limiting procedure as we have done before. We can apply Theorem 7.lc to
deduce that there exist points a e ~n and ~ e au, and a real number m 2: 1 such that
However, since hk (U) is a starlike domain with respect to 0, this implies that
a'[ a:: (a)] E hk(V), which leads to Df(a)a E h(~n). Since this contradicts the hypothesis we must have f(z) -< g(z). The function g is the solution of the differential equation
differential subordination. D
We close this section with two additional differential subordinations. The proofs will be omitted since their proofs are very similar to that of the last theorem.
