University of Illinois Urbana-Champaign

Growth comparisons for certain Nevanlinna theory functionals

Kwon, Ki-Ho

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https://hdl.handle.net/2142/22708

Description

Title
Growth comparisons for certain Nevanlinna theory functionals
Author(s)
Kwon, Ki-Ho
Issue Date
1991
Doctoral Committee Chair(s)
Miles, Joseph B.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
  • We show that for all entire f with $\vert{\rm f}(0)\vert\ge 1$ and all ${\rm r}>0$, $$\rm log\ M(r,f)\le d\sb\alpha T(r,f)\sp{1\over 2}T(\alpha r,f)\sp{1\over 2},\leqno(*)$$and$$\rm m\sbsp{p}{+}(r,f)\le d\sbsp{\alpha}{p-1\over p}\ T(r,f)\sp{p+1\over 2p}\ T(\alpha r,f)\sp{p-1\over 2p},$$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$$\rm d\sb\alpha = {4\sqrt{3}\ \alpha\sp{1\over 2}(\alpha\sp{1\over 2} + 1)\over \alpha - 1}.$$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\sb\alpha = o\left({1\over \alpha - 1}\right),\quad \alpha\to 1.$$
  • Suppose now that f(z) is a nonconstant meromorphic function in the plane, that m$\sb2$(r,f) is the L$\sb2$ norm of $\rm log\vert f(re\sp{i\theta})\vert,$ and that $\varphi$(x) is a positive increasing function satisfying $\rm \int\sp\infty{dx\over \varphi(x)} < \infty.$ Then we prove that there exists a set F with finite logarithmic measure such that$$\rm{\lim\limits\sb{r\to\infty\atop r\notin F}}\ {m\sb2 (r,f) \over T(r,f) \lbrack\varphi (log\ T(r,f))\rbrack\sp{1\over 2}} = 0.\leqno(**)$$The relation $(**)$ is shown to be sharp. We also prove several other theorems of $\rm {m\sb2(r,f)\over T(r,f)},$ and study the upper bounds for $\rm log\ M(r,f)\over T(r,f)$ for analytic functions on the unit disc.
Type of Resource
text
Permalink
http://hdl.handle.net/2142/22708
Copyright and License Information
Copyright 1991 Kwon, Ki-Ho

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