Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A sharp form of the Ahlfors' distortion theorem, with applications


Author: D. H. Hamilton
Journal: Trans. Amer. Math. Soc. 282 (1984), 799-806
MSC: Primary 30C50
DOI: https://doi.org/10.1090/S0002-9947-1984-0732121-3
MathSciNet review: 732121
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The constant appearing in the asymptotic version of the Ahlfors' distortion theorem is $ 1$. Also it is shown that for mean $ 1$-valent functions $ f = z + {a_2}{z^2} \cdots \left\Vert {{a_{n + 1}}\vert - \vert{a_n}} \right\Vert \leqslant 1$ for $ n \geqslant N(f)$.


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

  • [1] D. Aharanov and U. Srebro, Boundary behaviour of conformal and quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1982), 3-42. MR 698842 (84d:30029)
  • [2] L. Ahlfors, Untersuchungen zur Theorie der konformen Abbildung und der gonzen Functionen, Acta Soc. Sci. Fenn. Nova Ser. A 1 (1930), no. 9.
  • [3] B. G. Eke, Remarks on Ahlfors' distortion theorem, J. Analyse Math. 19 (1967), 97-134. MR 0215971 (35:6806)
  • [4] -, The asymptotic behaviour of areally mean $ p$-valent functions, J. Analyse Math. 20 (1967), 147-212. MR 0222279 (36:5331)
  • [5] D. H. Hamilton, The successive coefficients of univalent functions, J. London Math. Soc. (2) 25 (1982), 122-128. MR 645870 (83i:30009)
  • [6] W. K. Hayman, On the successive coefficients of univalent functions, J. London Math. Soc. (2) 38 (1963), 228-243. MR 0148885 (26:6382)
  • [7] -, Tauberian theorems for multivalent functions, Acta Math. 125 (1970), 269-298. MR 0268372 (42:3270)
  • [8] J. A. Jenkins and K. Oikawa, On the growth of slowly increasing unbounded harmonic functions, Acta Math. 124 (1970), 37-63. MR 0259147 (41:3789)
  • [9] K. W. Lucas, A two point bound for areally mean $ p$-valent functions, J. London Math. Soc. (2) 43 (1968),487-494. MR 0225997 (37:1587)
  • [10] Ch. Pommerenke, On the Hankel determinants of univalent functions, Mathematika 14 (1967), 108-112. MR 0215976 (35:6811)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30C50

Retrieve articles in all journals with MSC: 30C50


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0732121-3
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society