On the growth of meromorphic solutions of some higher order differential equations (Q1084240)

Let P be a polynomial of \(w,w',...,w^{(n)}\) (n\(\geq 1)\) with meromorphic coefficients: \(P=\Sigma c(z)w^{i_ 0}(w')^{i_ 1}...(w^{(n)})^{i_ n}\) and \(\Delta =\max \{i_ 0+2i_ 1+...+(n+1)i_ n\}\). We consider the differential equation (*) \(P^ m=\sum^{p}_{j=0}a_ jw^ j\) (0\(\leq p\leq m\Delta)\). The purpose of this paper is to estimate the Nevanlinna characteristic T(r,w) of meromorphic solutions \(w=w(z)\) in the plane of (*). In this paper, we give two theorems in general case and as a special case when \(P=(w^{(n)})^ m\), we give three theorems. For example, (1) When \(0\leq p\leq m-1\), the equation (*) has no admissible solutions except the following form: \(P^ m=a(w+b)^ p\). (2) When \(s\leq m-3\), \((w^{(n)})^ m=aw^ m+\sum^{s}_{j=0}a_ jw^ j\) has not admissible solutions.