On the growth of a composition of entire functions
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 that for $$\lim\limits_{r\to+\infty} \frac{\ln\ln\,M_{f(g)}(r)}{\ln\ln\,M_f(\exp\{\gamma(r)\})}=+\infty, \quad M...
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| Format: | Article |
| Language: | English |
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Vasyl Stefanyk Precarpathian National University
2018-01-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
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| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/1462 |
| Summary: | Let $\gamma$ be a positive continuous on $[0,\,+\infty)$ function increasing to $+\infty$ and $f$ and $g$ be arbitrary entire functions of positive lower order and finite order.
In order that for $$\lim\limits_{r\to+\infty} \frac{\ln\ln\,M_{f(g)}(r)}{\ln\ln\,M_f(\exp\{\gamma(r)\})}=+\infty, \quad M_f(r)=\max\{|f(z)|:\,|z|=r\}, $$ it is necessary and sufficient that $(\ln\,\gamma(r))/(\ln\,r)\to 0$ as $r\to+\infty$. This statement is an answer to the question posed by A.P. Singh and M.S. Baloria in 1991.
Also in order that $$ \lim\limits_{r\to+\infty}\frac{\ln\ln\,M_F(r)} {\ln\ln\,M_f(\exp\{\gamma(r)\})}=0,\quad F(z)=f(g(z)), $$ it is necessary and sufficient that $(\ln\,\gamma(r))/(\ln\,r)\to \infty$ as $r\to+\infty$. |
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| ISSN: | 2075-9827 2313-0210 |