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)



On uniqueness sets for areally mean $ p$-valent functions

Author: Enrique Villamor
Journal: Proc. Amer. Math. Soc. 122 (1994), 1143-1151
MSC: Primary 30C45
MathSciNet review: 1211592
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the sets of uniqueness of areally mean p-valent functions in the unit disc. Namely, if $ f(z)$ is in this class and has the same angular limit in a set E on the boundary of the unit disc, we prove that if p is small compared to the size of E then $ f(z)$ is constant. We then construct an areally mean p-valent function which shows that some condition on the size of the set E must be imposed.

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

  • [1] A. Beurling, Ensembles exceptionnels, Acta Math. 72 (1940), 1-13. MR 0001370 (1:226a)
  • [2] L. Carleson, Sets of uniqueness for functions analytic in the unit disc, Acta Math. 87 (1952), 325-345. MR 0050011 (14:261a)
  • [3] E. F. Collingwood and A. J. Lohwater, Theory of cluster sets, University Press, London, 1966. MR 0231999 (38:325)
  • [4] B. G. Eke, The asymptotic behavior of areally mean p-valent functions, J. Anal. Math. 20 (1967), 147-212. MR 0222279 (36:5331)
  • [5] W. K. Hayman, Multivalent functions, Cambridge Univ. Press, London, 1967. MR 1310776 (96f:30003)
  • [6] -, Private communication.
  • [7] O. Lehto and K. I. Virtanen, Boundary behavior and normal meromorphic functions, Acta Math. 97 (1957), 47-65. MR 0087746 (19:403f)
  • [8] Ch. Pommerenke, On normal and automorphic functions, Michigan Math. J. 21 (1974), 193-202. MR 0357779 (50:10246)
  • [9] M. Tsuji, Beurling's theorem on exceptional sets, Tôhoku Math. J. (2) 2 (1950), 113-125. MR 0040435 (12:692b)
  • [10] W. K. Hayman, Subharmonic functions, Vol. 2, Academic Press, London, 1990. MR 1049148 (91f:31001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30C45

Retrieve articles in all journals with MSC: 30C45

Additional Information

Keywords: Areally mean p-valent, logarithmic capacity
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society