ABSTRACT
LEMMA 8.3k. Let p e ..J/[~] and let q e Q(~). /fp is not subordinate to q, then there exist points z0 = x0 + iy0 e ~ and ~0 e d~ \E(q), and a k ~ 0 such that
(iv)
p'(z0 ) = kq'(~0 ) and Im p"(zo) ~ k 2 Im q"(~o)
and so we can define
for which p(~Yo) c q(~) and p(Lly 0
) ct. q(~). Since p(~y0 ) c q(~), there exists z0 e ()~Yo such that p(z0 ) e dq(~) . This implies that there
exists ~0 E a~ \E(q) such that p(zo) = q(~o). Thus the first two conditions of the lemma have been proved. The remaining two conditions
follow by applying Lemma 8.3j . 0
With these lemmas completed we can move on to the differential subordination problem. Given a set Q and function q, we are interested in determining functions 'I' for which
'¥.6[Q,q], consists of those functions 'Jf:C 3xU~C that satisfy the admissibility condition
t 2 q"(t;) Im q'(t;) ~ k 1m q'(t;), z E .6., t; E dd \E(q) and k ~ 0 .
