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Conformality and Semiconformality of a Function Holomorphic in the Disk

Shinji Yamashita
Transactions of the American Mathematical Society
Vol. 245 (Nov., 1978), pp. 119-138
DOI: 10.2307/1998859
Stable URL: http://www.jstor.org/stable/1998859
Page Count: 20
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Conformality and Semiconformality of a Function Holomorphic in the Disk
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Abstract

Conformality and semiconformality at a boundary point, of a function $f$ nonconstant and holomorphic in $|z| < 1$ are local properties. Therefore one would suspect the requirement of such global conditions on $f$ as $f$ is univalent in $|z| < 1$, or $f$ is a member of a larger class which contains all univalent functions in $|z| < 1$. We shall prove some extensions and new results without any assumption on $f$, or with a local assumption on $f$ at most. Our methods are, for the most part, different from the ones in the classical cases. One of the main tools is Theorem 8 on the angular limits of the real part of a holomorphic function and its derivative.

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