Abstract
Let / be analytic in D = {z : |z| < 1} with /(0) = /'(0) - 1 = 0. For y > 0, the largest a(y) and P(y)
are found such that
z/"(z)
/l+z\ t t
zf'(z) (\+jY
/(£)
(l±±Y
+ f'(z) <\l-z)
/(z) ^ 1 - z J
z <\\-z)
•
The results solve the inclusion problem for convex and starlike functions defined in a sector.
1991 Mathematics subject classification (Amer. Math. Soc): 30C45.
Results & Lemmas (2)
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THEOREM 1.
THEOREM 1. Let f be analytic in D with /(0) = /'(O) - 1 = 0 and 0 < p < 1. Then for z e D, |; f'(z) -z implies (1) where (2) tttf) = 1…
THEOREM 1. Let f be analytic in D with /(0) = /'(O) - 1 = 0 and 0 < p < 1. Then for z e D, | ; f'(z) \l-z implies (1) where (2) tttf) = 1 arctan Also for y > 0, (3) zf'iz)
THEOREM 2.
THEOREM 2. For y > 0, C S*(/?(y)) C where a(/ ) anrf ^(y) are given by (2) a/w/ (5) respectively. Furthermore, a(P(y)) and P(y) are the…
THEOREM 2. For y > 0, C S*(/?(y)) C where a(/}) anrf ^(y) are given by (2) a/w/ (5) respectively. Furthermore, a(P(y)) and P(y) are the largest numbers such that the inclusion holds. References [1] L. Brickman, D. J. Hallenbeck, T. M. Macgregor and D. R. Wilken, 'Convex hulls and extreme points of families of starlike and convex mappings', Trans. Amer. Math. Soc. 185 (1973), 413^t28. [2] T. H. Macgregor, 'A subordination for convex functions of order a', J. London. Math. Soc. 9 (1975), 530-536.
Function classes studied:
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