Length estimates for holomorphic functions

Abstract:

Let $f(z) = {a_0} + \Sigma _{k = n}^\infty {a_k}{z^k}(n \geqslant 1)$ be holomorphic in $U:\left | z \right | < 1$. MacGregor [1, Theorem 2] proved that if $l(r,f)$ is the length of the outer boundary of the image $D(r,f)$ of the disk: $\left | z \right | < r$ by $f$, then $2\pi {r^n}\left | {{a_n}} \right | \leqslant l(r,f)$ for $0 < r < 1$. We introduce the notion of the exact outer boundary ${C^\# }(r,f)$ of $D(r,f)$ and prove that $2\pi {r^n}\left | {{a_n}} \right | \leqslant {l^\# }(r,f) \leqslant l(r,f)$ for $0 < r < 1$, where ${l^\# }(r,f)$ is the length of ${C^\# }(r,f)$. We shall make use of the estimate to obtain a criterion for $f$ to be Bloch in $U$.