Regular functions $f(z)$ for which $zf^{’} (z)$ is $\alpha$-spiral-like

Abstract:

Let $\mathfrak {F}_\alpha ^\lambda$ be the class of functions $f(z) = z + {a_2}{z^2} + \cdots$ which are regular in $E = \{ z/|z| < 1\}$ and satisfy \[ \operatorname {Re} \{ {e^{i\alpha }}(1 + zf''(z)/f’(z))\} > \lambda \cos \alpha \] for some $\alpha ,|\alpha | < \pi /2$, and for some $\lambda ,0 \leq \lambda < 1$. The author finds a range on $\alpha$ for which $f(z)$ in $\mathfrak {F}_\alpha ^\lambda$ is univalent in $E$. In particular, the author improves upon the range on a for which $f(z) \in \mathfrak {F}_\alpha ^0$ is known to be univalent in $E$. Also a corresponding result is obtained for those functions $f(z)$ in $\mathfrak {F}_\alpha ^\lambda$ for which $f''(0) = 0$.