Analytic functions with large sets of Fatou points
HTML articles powered by AMS MathViewer
- by J. S. Hwang and Peter Lappan PDF
- Proc. Amer. Math. Soc. 90 (1984), 293-298 Request permission
Abstract:
For a function $f$ analytic in the unit disc $D$, and for each $\lambda > 0$, let $L\left ( \lambda \right ) = \left \{ {z \in D:\left | {f\left ( z \right )} \right | = \lambda } \right \}$ denote a level set for $f$. We introduce a class $\mathcal {L}$, of functions characterized by geometric properties of a collection of sets $\left \{ {L\left ( {{\lambda _n}} \right )} \right \}$, where $\left \{ {{\lambda _n}} \right \}$ is an unbounded sequence. We show that ${\mathcal {L}_1}$, is a proper subclass of the class $\mathcal {L}$ of G. R. MacLane. Let ${A_\infty }$ denote the set of points ${e^{i\theta }}$ at which the function $f$ has $\infty$ as an asymptotic value, and let $F\left ( f \right )$ denote the set of Fatou points of $f$. We prove that for a function $f$ in the class ${\mathcal {L}_1}$, if $\Gamma$ is an arc of the unit circle such that $\Gamma \cap {A_\infty } = \emptyset$, then almost every point of $\Gamma$ belongs to $F\left ( f \right )$.References
- N. U. Arakeljan, Uniform and tangential approximations by analytic functions, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 3 (1968), no. 4-5, 273–286 (Russian, with Armenian and English summaries). MR 0274770
- F. Bagemihl, Curvilinear cluster sets of arbitrary functions, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 379–382. MR 69888, DOI 10.1073/pnas.41.6.379
- F. Bagemihl, P. Erdös, and W. Seidel, Sur quelques propriétés frontières des fonctions holomorphes définies par certains produits dans le cercle-unité, Ann. Sci. Ecole Norm. Sup. (3) 70 (1953), 135–147 (French). MR 0058705
- Leon Brown, P. M. Gauthier, and W. Seidel, Complex approximation for vector-valued functions with an application to boundary behaviour, Trans. Amer. Math. Soc. 191 (1974), 149–163. MR 342707, DOI 10.1090/S0002-9947-1974-0342707-X
- E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge University Press, Cambridge, 1966. MR 0231999
- Stephen Dragosh, The distribution of Fatou points of bounded and normal analytic functions, Ann. Acad. Sci. Fenn. Ser. A. I. 496 (1971), 11. MR 301203
- G. R. MacLane, Asymptotic values of holomorphic functions, Rice Univ. Stud. 49 (1963), no. 1, 83. MR 148923
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 90 (1984), 293-298
- MSC: Primary 30D40
- DOI: https://doi.org/10.1090/S0002-9939-1984-0727253-5
- MathSciNet review: 727253