On deviations from rational functions of entire functions of finite lower order
Volume 91 / 2007
Annales Polonici Mathematici 91 (2007), 161-177
MSC: Primary 30D35; Secondary 30D30.
DOI: 10.4064/ap91-2-6
Abstract
Let $f$ be a transcendental entire function of finite lower order, and let $q_\nu$ be rational functions. For $0<\gamma<\infty\,$ let $$ B(\gamma):=\cases{ {\displaystyle\frac{\pi\gamma}{\sin\pi\gamma}}&\hbox{if }\gamma\leq 0.5,\cr \pi\gamma&\hbox{if }\gamma>0.5.\cr} $$ We estimate the upper and lower logarithmic density of the set $$ \Bigl\{r:\sum_{1\leq\nu\leq k}\log^+\max_{|z|=r}|f(z)-q_\nu(z)|^{-1}< B(\gamma)T(r,f)\Bigr\}. $$
