It follows that limr_i f(re'e) = f(ete) exists almost everywhere. We let Es(F) denote the set of extreme points of s(F). In [1] it was proved that if F' is in the Nevanlinna ...
transactions of the american mathematical Volume
308, Number
1, July
society 1988
SUBORDINATION
FAMILIES AND EXTREME POINTS
YUSUF ABU-MUHANNA AND D. J. HALLENBECK ABSTRACT. Let s(F) denote the set of functions subordinate to a univalent function F in A the unit disk. Let Bo denote the set of functions (z)analytic in A satisfying \is an extreme point of Bo and |0(e,!)| < 1 for almost all t, then
J2" log X(F((elt)[< 1 for almost all t, and ip is an inner function
such that [ip(0)\ ^ 1, then
[lit
(30)
/
Jo
logX(F(qb(elt)iP(eie)),dD)dt
= -oc
for almost all 8. PROOF. We just prove that under the assumption that F is a bounded univalent function in A then
(31)
/ \l Jo
- |0(e!t)|2)1/2|F'(0(el()iA(e'e))|dt
< +oo
for almost all 9. The proof of (30) follows from (31) in the way that Theorem 6 was deduced from Theorems 4 and 5. We note that Fatou's lemma gives
