On the zeros of certain homogeneous differential polynomials (Q1805499)

Let \(f\) be a meromorphic function of finite order and let \(a\) be a complex number satisfying \(a\neq 1\) and \(a\neq (n+ 1)/n\) for all natural numbers \(n\). It is shown that if \(f(z) f''(z)- af'(z)^ 2\) has only finitely many zeros, then \(f\) has the form \(f(z)= R(z) \exp P(z)\) with a rational function \(R\) and a polynomial \(P\). If \(f(z) f''(z)- af'(z)^ 2\) has no zeros and if \(f\) is transcendental, then \(f\) has the form \(f(z)= \exp(\alpha z+ \beta)\) with constants \(\alpha\) and \(\beta\). The result is connected to earlier work of Hayman, Mues and Langley dealing with the case that \(a= 0\) or that \(f\) is entire.