Julia sets of two permutable entire functions

Abstract

In this paper first we prove that if $f$ and $g$ are two permutable transcendental entire functions satisfying $f=f_1\left(h\right)$ and $g=g_1\left(h\right)$, for some transcendental entire function $h$, rational function $f_1$ and a function $g_1$, which is analytic in the range of $h$, then $F\left(g\right)\subset F\left(f\right)$. Then as an application of this result, we show that if $f\left(z\right)=p\left(z\right)e^{q\left(z\right)}+c$, where $c$ is a constant, $p$ a nonzero polynomial and $q$ a nonconstant polynomial, or $f\left(z\right)={\int }^{z}p\left(z\right)e^{q\left(z\right)}dz$, where $p,$$q$ are nonconstant polynomials, such that $f\left(g\right)=g\left(f\right)$ for a nonconstant entire function $g$, then $J\left(f\right)=J\left(g\right)$.