Growth comparisons for certain Nevanlinna theory functionals

We show that for all entire f with $\vert{\rm f}(0)\vert\ge 1$ and all ${\rm r}>0$, $\rm log M(r,f)\le d\sbα T(r,f)\sp{1\over 2}T(α r,f)\sp{1\over 2},\leqno(*)$and$\rm m\sbsp{p}{+}(r,f)\le d\sbsp{α}{p-1\over p} T(r,f)\sp{p+1\over 2p} T(α r,f)\sp{p-1\over 2p},$where $\alpha > 1,$ $\rm m\sbsp{p}{+}(r,f)$ is the L$\sb{\rm p}$ norm of $\rm log\sp+\vert f(re\sp{i\theta})\vert,$ and$\rm d\sbα = {4\sqrt{3} α\sp{1\over 2}(α\sp{1\over 2} + 1)\over α - 1}.$The inequality $(*)$ improves the well-known inequality $\rm log\ M(r,f)\le {R + r\over R - r}\ T(R,f),$ $\rm 0 0.$$Using a technique introduced by W. K. Hayman, we show all these inequalities are sharp, and in particular that $(*)$ does not hold in general for any d$\sb\alpha$ for which$\rm d\sbα = o\left({1\over α - 1}\right),\quad α\to 1.$

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Kwon, Ki-Ho

Contributors dc:contributor

• Miles, Joseph B.

• Mathematics

Statement dc:rights

• Copyright 1991 Kwon, Ki-Ho

Language dc:language

eng

Identifier

AAI9210882
(UMI)AAI9210882

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/22708