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)

 
 

 

Coefficients and normal functions


Author: Peter Lappan
Journal: Proc. Amer. Math. Soc. 85 (1982), 335-341
MSC: Primary 30D45
DOI: https://doi.org/10.1090/S0002-9939-1982-0656097-6
MathSciNet review: 656097
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f(z) = \sum {a_n}{z^n}$ be an analytic function in the unit disc. It is proved that if $ \{ {a_n}\} $ is a bounded monotone sequence of real numbers, or if $ \sum \vert{a_n} - {a_{n - 1}}\vert < \infty $ and $ {a_n} \nrightarrow 0$, then $ f(z)$ is a normal function. Examples are given to show that these results are delicate.


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

  • [1] J. M. Anderson, J. Clunie and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12-37. MR 0361090 (50:13536)
  • [2] F. Bagemihl and W. Seidel, Sequential and continuous limits of meromorphic functions, Ann. Acad. Sci. Fenn. Ser. AI 280 (1960), 1-17. MR 0121488 (22:12226)
  • [3] D. M. Campbell and G. Piranian, Problems on the normality of holomorphic functions, Houston J. Math. (to appear). MR 666140 (83i:30046)
  • [4] O. Lehto and K. I. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math. 97 (1957), 47-65. MR 0087746 (19:403f)
  • [5] J. H. Mathews, Coefficients of uniformly normal-Bloch functions, Yokohama Math. J. 21 (1973), 27-31. MR 0325975 (48:4321)
  • [6] J. J. Neitzke, Coefficients of Bloch functions, Ph.D. Dissertation, Michigan State Univ., 1980.
  • [7] L. R. Sons and D. M. Campbell, Hadamard gap series and normal functions, Bull. London Math. Soc. 12 (1980), 115-118. MR 571732 (81d:30050)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30D45

Retrieve articles in all journals with MSC: 30D45


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0656097-6
Keywords: Normal function, nonnormal point
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society