Maximum and Average Valence of Meromorphic Functions

Aimo Hinkkanen; Joseph Miles

doi:10.1007/s40315-024-00533-8

Maximum and Average Valence of Meromorphic Functions

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

Abstract

If f is a meromorphic function from the complex plane C to the extended complex plane C¯, for r>0 let n(r) be the maximum number of solutions in {z:|z|≤r} of f(z)=a for a∈C¯, and let A(r, f) be the average number of such solutions. Using a technique introduced by Toppila, we exhibit a meromorphic function for which lim inf_r→∞n(r)/A(r,f)≥1.07328.

Original language	English (US)
Pages (from-to)	575-603
Number of pages	29
Journal	Computational Methods and Function Theory
Volume	24
Issue number	3
Early online date	Mar 30 2024
DOIs	https://doi.org/10.1007/s40315-024-00533-8
State	Published - Sep 2024
Externally published	Yes

Keywords

Meromorphic functions
Nevanlinna theory
Primary 30D35
Value distribution

ASJC Scopus subject areas

Analysis
Computational Theory and Mathematics
Applied Mathematics

Online availability

10.1007/s40315-024-00533-8

Library availability

Discover UIUC Full Text

Cite this

@article{a9e2694168da4f2fa3e5aa7f3873ff9a,

title = "Maximum and Average Valence of Meromorphic Functions",

abstract = "If f is a meromorphic function from the complex plane C to the extended complex plane C¯, for r>0 let n(r) be the maximum number of solutions in {z:|z|≤r} of f(z)=a for a∈C¯, and let A(r, f) be the average number of such solutions. Using a technique introduced by Toppila, we exhibit a meromorphic function for which lim infr→∞n(r)/A(r,f)≥1.07328.",

keywords = "Meromorphic functions, Nevanlinna theory, Primary 30D35, Value distribution",

author = "Aimo Hinkkanen and Joseph Miles",

note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.",

year = "2024",

month = sep,

doi = "10.1007/s40315-024-00533-8",

language = "English (US)",

volume = "24",

pages = "575--603",

journal = "Computational Methods and Function Theory",

issn = "1617-9447",

publisher = "Springer",

number = "3",

}

TY - JOUR

T1 - Maximum and Average Valence of Meromorphic Functions

AU - Hinkkanen, Aimo

AU - Miles, Joseph

N1 - Publisher Copyright: © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.

PY - 2024/9

Y1 - 2024/9

N2 - If f is a meromorphic function from the complex plane C to the extended complex plane C¯, for r>0 let n(r) be the maximum number of solutions in {z:|z|≤r} of f(z)=a for a∈C¯, and let A(r, f) be the average number of such solutions. Using a technique introduced by Toppila, we exhibit a meromorphic function for which lim infr→∞n(r)/A(r,f)≥1.07328.

AB - If f is a meromorphic function from the complex plane C to the extended complex plane C¯, for r>0 let n(r) be the maximum number of solutions in {z:|z|≤r} of f(z)=a for a∈C¯, and let A(r, f) be the average number of such solutions. Using a technique introduced by Toppila, we exhibit a meromorphic function for which lim infr→∞n(r)/A(r,f)≥1.07328.

KW - Meromorphic functions

KW - Nevanlinna theory

KW - Primary 30D35

KW - Value distribution

UR - http://www.scopus.com/inward/record.url?scp=85189018188&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85189018188&partnerID=8YFLogxK

U2 - 10.1007/s40315-024-00533-8

DO - 10.1007/s40315-024-00533-8

M3 - Article

AN - SCOPUS:85189018188

SN - 1617-9447

VL - 24

SP - 575

EP - 603

JO - Computational Methods and Function Theory

JF - Computational Methods and Function Theory

IS - 3

ER -

Maximum and Average Valence of Meromorphic Functions

Abstract

Keywords

ASJC Scopus subject areas

Online availability

Library availability

Related links

Fingerprint

Cite this