Starlike functions

Abstract

Let S ∗ [ α ] {\mathcal {S}^\ast }[\alpha ] denote the class of functions f ( z ) = z + ∑ n = 2 ∞ a n z n f(z) = z + \sum \nolimits _{n = 2}^\infty {{a_n}{z^n}} analytic in | z | > 1 |z| > 1 and for which | z f ′ ( z ) / f ( z ) − 1 | > 1 − α |zf’(z)/f(z) - 1| > 1 - \alpha for | z | > 1 |z| > 1 and α ∈ [ 0 , 1 ) \alpha \in [0,1) . Sharp results concerning coefficients, distortion, and the radius of convexity are obtained. Furthermore, it is shown that ∑ n = 2 ∞ [ ( n − α ) / ( 1 − α ) ] | a n | > 1 \sum \nolimits _{n = 2}^\infty {[(n - \alpha )/(1 - \alpha )]|{a_n}| > 1} is a sufficient condition for f ( z ) ∈ S ∗ [ α ] f(z) \in {\mathcal {S}^\ast }[\alpha ] .