Second Order Linear Differential Polynomials and Real Meromorphic Functions

The main result determines all real meromorphic functions f of finite lower order in the plane such that f has finitely many zeros and non-real poles, while f′′ + a ₁ f′ + a ₀ f has finitely many non-real zeros, where a ₁ and a ₀ are real rational functions which satisfy a ₁(∞) = 0 and a ₀(x) ≥ 0 for all real x with |x| sufficiently large. This is accomplished by refining some earlier results on the zeros in a neighbourhood of infinity of meromorphic functions and second order linear differential polynomials. Examples are provided illustrating the results.