Length and area estimates of the derivatives of bounded holomorphic functions

Author:
Shinji Yamashita

Journal:
Proc. Amer. Math. Soc. **88** (1983), 29-33

MSC:
Primary 30C50

DOI:
https://doi.org/10.1090/S0002-9939-1983-0691273-9

MathSciNet review:
691273

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Abstract: MacGregor [**1**] and Yamashita [**5**] proved the estimates of the coefficient of the Taylor expansion of nonconstant and holomorphic in in terms of the area of the image of by and the length of its outer or exact outer boundary. We shall consider some analogous estimates in terms of the non-Euclidean geometry for bounded, , in . For example, is strictly less than the non-Euclidean length of the boundary of the image of , the multiplicity not being counted.

**[1]**T. H. MacGregor,*Length and area estimates for analytic functions*, Michigan Math. J.**11**(1964), 317-320. MR**0171003 (30:1236)****[2]**-,*Translations of the image domains of analytic functions*, Proc. Amer. Math. Soc.**16**(1965), 1280-1286. MR**0194600 (33:2810)****[3]**W. K. Hayman,*Mullivalent functions*, Cambridge Univ. Press, London, 1967.**[4]**R. Osserman,*The isoperimetric inequality*, Bull. Amer. Math. Soc.**84**(1978), 1182-1238. MR**0500557 (58:18161)****[5]**S. Yamashita,*Length estimates for holomorphic functions*, Proc. Amer. Math. Soc.**81**(1981), 250-252. MR**593467 (83c:30018)**

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DOI:
https://doi.org/10.1090/S0002-9939-1983-0691273-9

Article copyright:
© Copyright 1983
American Mathematical Society