.A, then the following sharp implications hold:

Re [ f(z) I z r' 2 > 112. This inequality is sharp with extremal function f(z) = z/(1-zn)Zin.

(2.6-7)

where

(2.3-11) is satisfied. This follows immediately since

(2.6-8) Re \jf(ip, <1) = 1 + ?:_ -f-- ~ 1 - 1 = 0, n p + 1

We next prove a theorem which in its original form was proved in 1959 by

Sakaguchi [ 362 ]. Libera [ 167] and MacGregor [ 175] extended it into the

form presented here. They are among the many authors who have applied

the result to various problems.