On the growth of a composition of entire functions
Journal Title: Карпатські математичні публікації - Year 2017, Vol 9, Issue 2
Abstract
Let $\gamma$ be a positive continuous on $[0,\,+\infty)$ function increasing to $+\infty$ and $f$ and $g$ be arbitrary entire functions of positive lower order and finite order. In order to $$ \lim\limits_{r\to+\infty} \dfrac{\ln\ln\,M_{f(g)}(r)}{\ln\ln\,M_f(\exp\{\gamma(r)\})}=+\infty, \quad M_f(r)=\max\{|f(z)|:\,|z|=r\}, $$ it is necessary and sufficient $(\ln\,\gamma(r))/(\ln\,r)\to 0$ as $r\to+\infty$. This statement is an answer to the question posed by A.P. Singh and M.S. Baloria in 1991. Also in order to $$ \lim\limits_{r\to+\infty}\dfrac{\ln\ln\,M_F(r)} {\ln\ln\,M_f(\exp\{\gamma(r)\})}=0,\quad F(z)=f(g(z)), $$ it is necessary and sufficient $(\ln\,\gamma(r))/(\ln\,r)\to \infty$ as $r\to+\infty$.
Authors and Affiliations
M. Sheremeta
Keywords
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- DOI 10.15330/cmp.9.2.181-187
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