On the convergence of infinite compositions of entire functions (Q431199)

The author studies the convergence of infinite compositions of entire functions, proving the following result: Let \(f_n(z)=a_{n,0}+a_{n,1}z+a_{n,2}z^2+a_{n,3}z^3+\cdots\), \(n=1,2,3,\dots\), be entire functions and set \[ A_n=\sup\{|a_{n,r}|^{1/(r-1)}: r=2,3,4,\dots\}. \] Suppose that \[ \sum_{n=1}^{\infty}A_n,\quad \sum_{n=1}^{\infty}|a_{n,0}|,\quad \prod_{n=1}^{\infty}a_{n,1} \] are convergent. If we set \[ F_{N}(z)=f_1\circ f_2\circ \cdots\circ f_N(z) \] then the sequence of entire functions \(\{F_N(z)\}_{N=1}^{\infty}\) uniformly converges to an entire function on any compact subset of \(\mathbb{C}\).