The access theorem for subharmonic functions
HTML articles powered by AMS MathViewer
- by R. Hornblower and E. S. Thomas
- Trans. Amer. Math. Soc. 172 (1972), 287-297
- DOI: https://doi.org/10.1090/S0002-9947-1972-0308420-8
- PDF | Request permission
Abstract:
A chain from a point ${z_0}$ of the open unit disk $\Delta$ to the boundary of $\Delta$ is a set $\Gamma = \cup \{ {\gamma _n}|n = 0,1,2, \cdots \}$ where the ${\gamma _n}$ are compact, connected subsets of $\Delta ,{z_0}$ is in ${\gamma _0},{\gamma _n}$ meets ${\gamma _{n + 1}}$ and the ${\gamma _n}$ approach the boundary of $\Delta$. The following “Access Theorem” is proved: If $u$ is subharmonic in $\Delta ,{z_0}$ is a point of $\Delta$ and $M < u({z_0})$, then there is a chain from ${z_0}$ to the boundary of $\Delta$ on which $u \geq M$ and on which $u$ tends to a limit. A refinement, in which the chain is a polygonal arc, is established, and an example is constructed to show that the theorem fails if $M = u({z_0})$ even for bounded, continuous subharmonic functions.References
- Marcel Brelot, Étude des fonctions sous-harmoniques au voisinage d’un point singulier, Ann. Inst. Fourier (Grenoble) 1 (1949), 121–156 (1950) (French). MR 37416
- W. K. Hayman, On the characteristic of functions meromorphic in the unit disk and of their integrals, Acta Math. 112 (1964), 181–214. MR 168763, DOI 10.1007/BF02391770
- Maurice Heins, Selected topics in the classical theory of functions of a complex variable, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, 1962. MR 0162913
- John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
- R. J. M. Hornblower, Subharmonic analogues of MacLane’s classes, Ann. Polon. Math. 26 (1972), 135–146. MR 364642, DOI 10.4064/ap-26-2-135-146
- G. R. MacLane, Asymptotic values of holomorphic functions, Rice Univ. Stud. 49 (1963), no. 1, 83. MR 148923 M. N. M. Talpur, Thesis, London, 1967.
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 172 (1972), 287-297
- MSC: Primary 31A20
- DOI: https://doi.org/10.1090/S0002-9947-1972-0308420-8
- MathSciNet review: 0308420