Growth of meromorphic solutions of some functional equations I
The authors are concerned with meromorphic solutions of the functional equation \[ \sum^n_{j=0} a_j(z)f(c^jz) =Q(z) \] where \(Q\) and the \(a_j\) are polynomials without common zeros, \(a_n(z)a_0 (z)\neq 0\) and \(0<|c |<1\). They show that each transcendental meromorphic solution \(f(z)\) of this equation satisfies \[ m(r,f)= \sigma_f(\log r)^2 \bigl(1+o(1)\bigr) \] for some constant \(\sigma_f\). Here \(m(r,f)\) signifies the proximity function of \(f(z)\) (standard notations in the Nevanlinna theory).
scientific article; zbMATH DE number 1742892