Second order linear differential polynomials and real meromorphic functions
The author continues to study the value distribution of real meromorphic functions \(f\) and linear differential polynomials in \(f\). The author proved in [Proc. R. Soc. Edinb. Sect. A. 141, No. 3, 631--639 (2011; Zbl 1242.30022)] the following theorem: Let \(f\) be a real meromorphic function of finite lower order in the plane such that \(f\) has finitely many zeros and non-real poles. Let \(a_1\) and \(a_0\) be real rational functions such that \(a_1(z)\) and \(za_0(z)\) vanish at infinity, and assume that \(F=f''+ a_{1} f' + a_{0} f\) has finitely many non-real zeros. Then \(f=Se^P\) with \(S\) a rational function and \(P\) a polynomial. The main result in this paper is an extension of this theorem. The author refines some earlier results on zeros of meromorphic functions and second order linear differential polynomials in a neighbourhood of infinity. A function \(a\) is called rational at infinity if it is analytic in \(S\leq |z| < \infty\) for some \(S>0\) and has a pole or removable singularity at infinity. For such \(a\) the degree at infinity is \(\deg_\infty (a)=\lim_{|z|\to\infty} \log|a(z)|/\log |z|\). The author first shows that if a meromorphic function \(f\) and \(F=f''+ a_{1} f' + a_{0} f\) have finitely many zeros, where \(a_1\) and \(a_0\) are rational functions with \(a_1(\infty)=0\) and \(\deg_{\infty}(a_0)\) even and non-negative, then there are five possibilities. In the main result, by using the expressions of these possibilities, he 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 \(F=f''+ a_{1} f' + a_{0} f\) has finitely many non-real zeros, where \(a_1\) and \(a_0\) are real rational functions which satisfy \(a_{1}(\infty) = 0\) and \(a_{0}(x) \geq 0\) for all real \(x\) with \(|x|\) sufficiently large. The author gives several examples which illustrate the results mentioned above.
scientific article; zbMATH DE number 6135239
scientific article