On functions of bounded boundary rotation

Abstract:

Let $U = \{ z = r{e^{i\theta }}\left | {r < 1\} } \right .$. For $k \geqq 2$ let ${V_k}$ be the class of normalized analytic functions $f(z)$ such that the boundary rotation of $f(U)$ is at most $k\pi$. Let $A(r)$ be the integral \[ \int _0^{2\pi } {\int _0^r {\left | {f’(\rho {e^{i\theta }})} \right |} } {^2}\rho d\,\rho d\,\theta ,\] $L(r)$ the length of the image of the circle $\left | z \right | = r$ under the mapping $f(z)$. In this paper the author proves that for $z \in U$ if $f(z) \in {V_k}$ then \[ \limsup _{r \to 1} \left ( \sup _{f \in {V_k}} L(r) \right ) \left ( \pi A(r)\log \left ( \frac {1 + r}{1 - r} \right ) \right )^{-1/2} \leqq k. \] This generalizes to arbitrary $k \geqq 2$ the recent result of Nunokawa for the case $k = 2$.