Spherical derivative of meromorphic solutions of analytic differential equations (Q641974)

Let \[ \left({dw\over dz}\right)^n= \sum^m_{j=1} P_j(z,w)\,D_j(w), \] where \(P_j(z,w)= \sum^m_{i=0} a_{ij}(z)\,w^i\), \(1\leq j\leq m\), \(a_{ij}\) are analytic functions in the unit disk \(\mathbb{D}\), and \[ D_j(w)= \Biggl({dw\over dz}\Biggr)^{j_1} \Biggl({d^2w\over dz^2}\Biggr)^{j_2}\cdots \Biggl({d^kw\over dz^k}\Biggr)^{j_k},\;1\leq k\leq m. \] Let \(\nu_j= j_1+ 2j_2+\cdots+ kj_k\) and \(\nu(P)= \max_{1\leq j\leq m}\{\nu_j\}\). If \(f\) is a meromorphic solution of the above equation with \(n>\nu(P)\), conditions are given on the growth of the functions \(a_{ij}\) such that the growth of the spherical derivative of \(f\) can be estimated.