Distortions properties of alpha-starlike functions

Abstract:

Let $\alpha$ be real and suppose that $f(z) = z + \Sigma _2^\infty {a_n}{z^n}$ is regular in the unit disc $D$ with $f(z)f’(z) \ne 0$ in $0 < |z| < 1$. If $\operatorname {Re} [(1 - \alpha )zf’(z)/f(z) + \alpha ((zf''(z)/f’(z)) + 1)] > 0$ for $z \in D$, then $f(z)$ is said to be an alpha-starlike function. These functions are univalent and they very naturally unify the classes of starlike $(\alpha = 0)$ and convex $(\alpha = 1)$ functions. The author obtains the $\tfrac {1}{4}$-theorem, sharp bounds on $|f(z)|$ and $|f’(z)|$, and growth conditions on $M(r)$.