On meromorphic solutions of certain functional equations

TLDR: In this article, the authors considered the special case when ¢(z) is a non-linear polynomial, i.e., the equations f(p(z ) ) = q( f( z ) )+ h(z), where p (z ) = a p z P + a p l z P l +... + a o is a polynomial of degree p (p > l ).

Abstract: The purpose of this paper is to consider meromorphic solutions of the functional equations f(q~(z)) = q( f (z ) ) + h(z ) (1) f (~ ( z ) ) = g(z )q( f ( z ) ) , (2) where f ( f~ constant), g (g~0) , h are meromorphic functions (in the whole open plane), ~(z) is a non-constant entire function and q(z) is a rational [unction of order k (k >>-1). Most of our results will concern the special case, when ¢(z) is a non-linear polynomial, i.e. the equations f (p(z ) ) = q( f ( z ) )+ h(z) (3) f (p(z ) ) = g(z)q( f (z) ) (4) where p ( z ) = a p z P + a p _ l z P l + . . . + a o is a polynomial of degree p ( p > l ) . In what follows we shall assume, unless otherwise stated that f, g, h, q~, q, p denote such functions. Certain special cases of these equations have been considered before. Thus, the equation