Meromorphic solutions of some functional equations

Abstract

It is shown that transcendental meromorphic solutions f(z) of the functional equation ]C?=o a j( z )f( c ^z) = Q( z ) where 0 < |c| < 1 is a complex number and aj(Z), j = 0,1,..., n, and Q(z) are rational functions with ao(z) ^ 0, an(z) = 1, satisfy T(r,f) = 0((logr) 2 ) and (logr) 2 = 0{T(r,f)).Moreover, in the case n = 2 and Q(z) = 0, necessary and sufficient conditions for the existence of solutions are given.