Value distribution of meromorphic functions satisfying generalised second Painlevé differential equation (Q1282002)

This article is devoted to considering the differential equation \(f^{(2m)}=Kf^3+zf+C\), where \(m\geq 1\) is an integer, and \(K\) and \(C\) are complex constants. This equation may be understood as a generalized second Painlevé equation. It is easy to see that for any meromorphic solution \(f\), all poles of \(f\) are of multiplicity \(m\) and the deficiency \(\delta(a,f)=0\) for every \(a\in\mathbb{C}\). Moreover, counting a-points of \(f\) of multiplicity \(\geq s\) only, and applying for the integrated counting function \(N_s(r,a,f)\) of such a-points the reduced multiplicity \(\nu(a,z)-s\) instead of the multiplicity \(\nu(a,z)\), the following results will be proved: \[ \theta_1(\infty,f)= \liminf_{r\to\infty} {N_1(r,f) \over T(r,f)}= {m-1\over m},\quad \theta_m (a,f)=\liminf_{r\to\infty} {N_m(r,f) \over T(r,f)} \leq{m\over 2m+1}. \] Moreover, \[ \liminf_{r\to\infty} {N\left(r, {1\over f'} \right)\over T(r,f)}={m+1 \over m},\quad \liminf_{r\to \infty} {N\left(r, {1\over f'}\right) +2N(r,f)- N(r,f')\over T(r,f)}=2. \] The proofs apply the standard Nevalinna theory. Some confusion appears in the notations.