ABSTRACT
We will discuss a few special cases of the theorem. If we take n == A. = 1, then the unique solution of (3.1-9) is ~0 = ~0 (1, 1) = 1.21872 ··· , and we deduce that the largest value of a(~,1, 1) for the sector h(U) corresponds to a 0 = a(~0 ,1, 1) == 1.781···. In this case the angle of sector h(U) is 320.62 · · · o, while the angle of sector q(U) is 219.36 · · · 0 •
(3.1-11) p(z) + zp'(z) -< ! ~ ~ ~ p(z) -< [ ! ~ ~ r' where ~ == 0.638 · · · . [The angle of sector h(U) is 180°, while the angle of sector q(U) is 114.892 · · · 0 .] The differential subordination in this last case
coincides with Theorem 3.lb, which supplies the best dominant. Hence we
can replace (3.1-11) with the sharp result:
p(z) + zp'(z) -< 1 1
Mocanu, Ripeanu and Popovici [ 242] showed that in this case
We next return to Goluzin's differential subordination (3.1-2) and present an extension due to Suffridge. We supply a different proof employing
differential subordinations.
