Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Length estimates for holomorphic functions


Author: Shinji Yamashita
Journal: Proc. Amer. Math. Soc. 81 (1981), 250-252
MSC: Primary 30C99; Secondary 30D45
MathSciNet review: 593467
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f(z) = {a_0} + \Sigma _{k = n}^\infty {a_k}{z^k}(n \geqslant 1)$ be holomorphic in $ U:\left\vert z \right\vert < 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\vert z \right\vert < r$ by $ f$, then $ 2\pi {r^n}\left\vert {{a_n}} \right\vert \leqslant l(r,f)$ for $ 0 < r < 1$. We introduce the notion of the exact outer boundary $ {C^\char93 }(r,f)$ of $ D(r,f)$ and prove that $ 2\pi {r^n}\left\vert {{a_n}} \right\vert \leqslant {l^\char93 }(r,f) \leqslant l(r,f)$ for $ 0 < r < 1$, where $ {l^\char93 }(r,f)$ is the length of $ {C^\char93 }(r,f)$. We shall make use of the estimate to obtain a criterion for $ f$ to be Bloch in $ U$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30C99, 30D45

Retrieve articles in all journals with MSC: 30C99, 30D45


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0593467-8
Article copyright: © Copyright 1981 American Mathematical Society