On the growth of a composition of entire functions

Array

Authors

  • M.M. Sheremeta Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine

DOI:

https://doi.org/10.15330/cmp.9.2.181-187

Keywords:

entire function, composition of functions, generalized order

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 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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Published

2018-01-02

How to Cite

(1)
Sheremeta, M. On the Growth of a Composition of Entire Functions: Array. Carpathian Math. Publ. 2018, 9, 181-187.

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Section

Scientific articles