Results on meromorphic φ -normal functions

Rauno Aulaskari; Shamil Makhmutov; Jouni Rättyä

doi:10.1080/17476930902998860

Results on meromorphic φ -normal functions

Rauno Aulaskari^*, Shamil Makhmutov, Jouni Rättyä

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

Abstract

Let φ: [0, 1) → (0, ∞) be an increasing function. A meromorphic function f in the unit disc is said to be φ -normal if its spherical derivative f^#(z):=|f′(z)|/(1 +|f(z)|²) satisfies f^#(z) = O{script} (φ(|z|)) as |z| → 1^-. This article is devoted to the study of meromorphic φ-normal functions. In particular, an analogue of the Lohwater-Pommerenke theorem and several equivalent characterizations for φ-normal functions are established under certain regularity conditions on φ.

Original language	English
Pages (from-to)	855-863
Number of pages	9
Journal	Complex Variables and Elliptic Equations
Volume	54
Issue number	9
DOIs	https://doi.org/10.1080/17476930902998860
Publication status	Published - Sept 2009

Keywords

Bloch function
Chordal distance
Meromorphic function
Normal family
Normal function
Spherical derivative
Unit disc

ASJC Scopus subject areas

Analysis
Numerical Analysis
Computational Mathematics
Applied Mathematics

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10.1080/17476930902998860

Cite this

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title = "Results on meromorphic φ -normal functions",

abstract = "Let φ: [0, 1) → (0, ∞) be an increasing function. A meromorphic function f in the unit disc is said to be φ -normal if its spherical derivative f#(z):=|f′(z)|/(1 +|f(z)|2) satisfies f#(z) = O{script} (φ(|z|)) as |z| → 1-. This article is devoted to the study of meromorphic φ-normal functions. In particular, an analogue of the Lohwater-Pommerenke theorem and several equivalent characterizations for φ-normal functions are established under certain regularity conditions on φ.",

keywords = "Bloch function, Chordal distance, Meromorphic function, Normal family, Normal function, Spherical derivative, Unit disc",

author = "Rauno Aulaskari and Shamil Makhmutov and Jouni R{\"a}tty{\"a}",

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T1 - Results on meromorphic φ -normal functions

AU - Aulaskari, Rauno

AU - Makhmutov, Shamil

AU - Rättyä, Jouni

N1 - Funding Information: The authors would like to thank the referee for a very careful reading of the manuscript, and for pointing out several misprints. This research was supported in part by the Academy of

PY - 2009/9

Y1 - 2009/9

N2 - Let φ: [0, 1) → (0, ∞) be an increasing function. A meromorphic function f in the unit disc is said to be φ -normal if its spherical derivative f#(z):=|f′(z)|/(1 +|f(z)|2) satisfies f#(z) = O{script} (φ(|z|)) as |z| → 1-. This article is devoted to the study of meromorphic φ-normal functions. In particular, an analogue of the Lohwater-Pommerenke theorem and several equivalent characterizations for φ-normal functions are established under certain regularity conditions on φ.

AB - Let φ: [0, 1) → (0, ∞) be an increasing function. A meromorphic function f in the unit disc is said to be φ -normal if its spherical derivative f#(z):=|f′(z)|/(1 +|f(z)|2) satisfies f#(z) = O{script} (φ(|z|)) as |z| → 1-. This article is devoted to the study of meromorphic φ-normal functions. In particular, an analogue of the Lohwater-Pommerenke theorem and several equivalent characterizations for φ-normal functions are established under certain regularity conditions on φ.

KW - Bloch function

KW - Chordal distance

KW - Meromorphic function

KW - Normal family

KW - Normal function

KW - Spherical derivative

KW - Unit disc

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JF - Complex Variables and Elliptic Equations

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ER -

Results on meromorphic φ -normal functions

Abstract

Keywords

ASJC Scopus subject areas

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