Gelfer functions, integral means, bounded mean oscillation, and univalency

Author:
Shinji Yamashita

Journal:
Trans. Amer. Math. Soc. **321** (1990), 245-259

MSC:
Primary 30C45; Secondary 30C50

DOI:
https://doi.org/10.1090/S0002-9947-1990-1010891-X

MathSciNet review:
1010891

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Abstract: A Gelfer function is a holomorphic function in such that and for all , in . The family of Gelfer functions contains the family of holomorphic functions in with and Re in . If is holomorphic in and if the mean of on the circle is dominated by that of a function of as , then . This has two recent and seemingly different results as corollaries. A core of the proof is the fact that if . Besides the properties obtained concerning itself, we shall investigate some families of functions where the roles played by in Univalent Function Theory are replaced by those of . Some exact estimates are obtained.

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DOI:
https://doi.org/10.1090/S0002-9947-1990-1010891-X

Article copyright:
© Copyright 1990
American Mathematical Society