Julia sets of two permutable entire functions

摘要

In this paper first we prove that if f and g are two permutable transcendental entire functions satisfying f = f_1(h) and g = g_1(h), 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(g) is contained in F(f). Then as an application of this result, we show that if f(z) = p(z)e~(q(z))+c, where c is a constant, p a nonzero polynomial and q a nonconstant polynomial, or f(z) = ∫~z p(z)e~(q(z)) dz, where p,q are nonconstant polynomials, such that f(g) = g(f) for a nonconstant entire function g, then J(f) = J(g).