Zeros of derivatives of meromorphic functions
The first part of this paper is a survey type presentation of the topics considering the link between the poles of meromorphic functions and the zeros of their derivatives, while the last part is devoted to proving an improved version of one of the main results in this field. The first part offers several conjectures in the present topic describing some of the related key results. These conjectures include the Gol'dberg conjecture proposing that \(\overline{N}(r,f)\leq N\big(r,1/f^{(k)})+o(T(r,f)\big)\) n.e. holds for transcendental meromorphic functions with \(k\geq 2\), and the Wiman-Valiron conjecture claiming that a real entire function \(f\) belongs to the Laguerre-Pólya class \(LP\) provided that \(f,f''\) both have only real zeros. In particular, a number of results related to the Wiman-Valiron conjecture from the last fifty years are recalled. Some of these results may be combined to the following theorem: For a real entire function \(f\), the number of non-real zeros of \(f^{(k)}\) tends to a limit as \(k\rightarrow\infty\), the limit being either zero of \(\infty\). The paper then continues by recalling some results due to \textit{S. Hellerstein, L.-C. Shen, J. Williamson} and \textit{J. Rossi}, see e.g. [\textit{S. Hellerstein, L.-C. Shen} and \textit{J. Williamson,} Trans. Am. Math. Soc. 285, 759--776 (1984; Zbl 0549.30021)], presenting the meromorphic analogue of the Wiman--Pólya topic. In fact, here \(f\) is supposed to be meromorphic non-entire such that some of the derivatives of \(f\) have only real zeros, asking whether \(f\) can be determined explicitly. Essentially improving one of his previous results (to appear in Proc. Amer. Math. Soc.), the author then provides a proof for the following result: Let \(f\) be a real meromorphic function, not of the form \(f=Se^{P}\) with \(S\) rational and \(P\) polynomial, and let \(\mu ,k\) be integers such that \(1\leq\mu <k\). Moreover, assume that all but finitely many zeros of \(f, f^{(k)}\) are real, and that \(f^{(\mu )}\) has finitely many zeros. Then \(\mu =1\), \(k=2\) and \(f\) is of the form \[ f(z)=\frac{R(z)e^{icz}-1}{AR(z)e^{icz}-\overline{A}}, \] where \(c\in (0,\infty )\), \(A\in\mathbb{C}\setminus\mathbb{R}\) and \(R(z)\) is a rational function with \(|R(x)|=1\) whenever \(x\) is real. In addition, all but finitely many poles of \(f\) must be real. This result in fact suggests that at least for non-entire meromorphic functions, the second derivative has a somewhat different character compared to the higher derivatives. The non-trivial proof of this last result is based on proving that first \(f'/f\) and then \(f\) itself are of finite order of growth. After this is established, the final argument then follows by using standard Gundersen estimates for logarithmic derivatives.
scientific article; zbMATH DE number 5840918
scientific article