An integral criterion for normal functions
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- by Rauno Aulaskari and Peter Lappan
- Proc. Amer. Math. Soc. 103 (1988), 438-440
- DOI: https://doi.org/10.1090/S0002-9939-1988-0943062-X
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Abstract:
A new characterization for normal functions is given. It is shown that a function $f$ meromorphic in the unit disk is a normal function if and only if for each $\delta > 0$ and each $p > 2$ there exists a constant ${K_f}(\delta ,p)$ such that, for each hyperbolic disk $\Omega$ with hyperbolic radius $\delta$, \[ \iint _\Omega {{{(1 - {{\left | z \right |}^2})}^{p - 2}}{{({f^ \ne }(z))}^p}dA(z) \leq {K_f}(\delta ,p)},\] where ${f^ \ne }(z)$ denotes the spherical derivative of $f$ and $dA(z)$ is the Euclidean element of area. It is shown by example that this characterization is not valid for $p = 2$.References
- Paul Gauthier, A criterion for normalcy, Nagoya Math. J. 32 (1968), 277–282. MR 230891
- V. I. Gavrilov, On the distribution of values of functions meromorphic in the unit circle, which are not normal, Mat. Sb. (N.S.) 67 (109) (1965), 408–427. MR 0197734
- Peter Lappan, A non-normal locally uniformly univalent function, Bull. London Math. Soc. 5 (1973), 291–294. MR 330467, DOI 10.1112/blms/5.3.291
- Peter Lappan, A criterion for a meromorphic function to be normal, Comment. Math. Helv. 49 (1974), 492–495. MR 379850, DOI 10.1007/BF02566744
- Olli Lehto and K. I. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math. 97 (1957), 47–65. MR 87746, DOI 10.1007/BF02392392
- A. J. Lohwater and Ch. Pommerenke, On normal meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A. I. 550 (1973), 12. MR 338381
- Shinji Yamashita, Criteria for functions to be Bloch, Bull. Austral. Math. Soc. 21 (1980), no. 2, 223–227. MR 574841, DOI 10.1017/S0004972700006043
Bibliographic Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 103 (1988), 438-440
- MSC: Primary 30D45
- DOI: https://doi.org/10.1090/S0002-9939-1988-0943062-X
- MathSciNet review: 943062