On the zeros of certain homogeneous differential polynomials

Abstract

E. Mues [10] proved in 1978 that if a ∈ C\{1} and if f is a transcendental entire function which is not of the form f(z) = exp(αz + β) where α, β ∈ C, then f(z)f ′′(z) − af ′(z)2 has at least one zero. The case a = 0 is due to W. K. Hayman [5, Theorem 5]. As shown by examples like f(z) = cos z the conclusion need not hold if a = 1, see [10] for further examples. If we allow f to be meromorphic, then we have further exceptional values for a. In fact, an easy computation shows that if a = (n + 1)/n for some n ∈ N and if f(z) = F (z)−n for an entire function F with the the property that F ′′ has no zeros, then f(z)f ′′(z)−af ′(z)2 = −nF (z)−2n−1F ′′(z) has no zeros. It seems reasonable to conjecture if f is a transcendental meromorphic function not of the form f(z) = exp(αz + β) and if a 6= 1 and a 6= (n + 1)/n, then f(z)f ′′(z) − af ′(z)2 has at least one zero. This has recently been proved by J. K. Langley [8] in the case that a = 0 and had been obtained earlier by Mues [9] in this case for functions of finite lower order.