On differential polynomials, fixpoints and critical values of meromorphic functions (Q1297008)

Using the Nevanlinna theory and the theory of harmonic measure, the author studies meromorphic functions \(f\) such that \(ff'' -{\alpha f'}^2\) has few zeros. In particular, the following result is obtained. Suppose that \[ N(r,{1\over ff'' -{\alpha f'}^2})=o\Biggl(T\biggl(r,{f\over f'}\biggr)\Biggr) \] where \(f\) is a transcendental meromorphic function, \(f'/f\) has finite lower order, \(\alpha\neq 1\), \({1\over \alpha-1}\) is not a positive integer. Then \(f(z)=\exp(Az+B)\). Further the case \(\alpha=1\) is considered. Among the other results, the following is proved. Suppose \(T(r,f)=O(r(\ln r)^\delta)\) where \(\delta>0\), \(e^8(200\delta)^{1/2}<1/4\), \(f\) is a transcendental meromorphic function such that \(f''\) has finitely many zeros. Then \(f\) has finitely many poles.