On a subclass of spiral-like functions

Abstract:

Let $\alpha \geqq 0,0 \leqq \beta < 1,|\lambda | < \pi /2$ and suppose that $f(z) = z + \sum \nolimits _{n = 2}^\infty {{a_n}{z^n}}$ is holomorphic in $U = \{ z:|z| < 1\}$. If \[ \operatorname {Re} \left [ {{e^{i\lambda }}\frac {{zf’(z)}}{{f(z)}} + \alpha \left ( {\frac {{zf''(z)}}{{f’(z)}} + 1 - \frac {{zf’(z)}}{{f(z)}}} \right )} \right ] > \beta \cos \lambda \] for $z \in U$, then $f(z)$ is said to be $\alpha$ - $\lambda$-spiral-like of order $\beta$ and we write $f(z) \in S_\alpha ^\lambda (\beta )$. The author shows that for each $\alpha \geqq 0, \alpha$ - $\lambda$ -spiral-like functions of order $\beta$ are $\lambda$-spiral-like of order $\beta$. The following representation theorem is obtained: The function $f(z) \in S_\alpha ^\lambda (\beta )(\alpha > 0,0 \leqq \beta < 1,|\lambda | < \pi /2)$, if and only if there exists a function $F(\zeta )\lambda$-spiral-like of order $\beta$ such that \[ F(z) = {\left [ {({e^{i\lambda }}/\alpha )\int _0^z {F{{(\zeta )}^{{e^{i\lambda }}/\alpha }}{\zeta ^{ - 1}}d\zeta } } \right ]^{\alpha {e^{ - i\lambda }}}}.\] A distortion theorem for $\log |f(z)/z|$ and a rotation theorem for $\arg f(z)/z$ are also proved for functions $f(z) \in S_0^\lambda (\beta )$.