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_1f′ + a_0f 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 {pipe}x{pipe} 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.
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