# On the growth of a composition of entire functions

• M.M. Sheremeta Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine
Keywords: entire function, composition of functions, generalized order
Published online: 2018-01-02

### 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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How to Cite
(1)
Sheremeta M. On the Growth of a Composition of Entire Functions. Carpathian Math. Publ. 2018, 9 (2), 181-187.