Zeros of derivatives of real meromorphic functions
The question of characterizing those real entire functions which together with some derivative have only real zeros, or only finitely many non-real zeros, has a long history, much of which has focussed on conjectures by Pólya and Wiman. By now, many questions in this area have been answered, in particular the conjectures by Pólya and Wiman were confirmed. Much less is known for meromorphic functions. One of the results obtained here says that if \(f\) is a real meromorphic function and \(k\geq 2\), then \(f\) has the form \(f=Se^P\) with a rational function \(S\) and a polynomial \(P\) if the following four conditions are satisfied: (i) \(f\) has finitely many non-real zeros; (ii) \(f^{(k)}\) has finitely many zeros; (iii) there exists \(M\) such that the multiplicity \(m_0\) of each pole \(z_0\) satisfies \(m_0\leq M+|z_0|^M\); (iv) if \(k=2\), then \(f'/f\) has finite lower order or the non-real poles of \(f\) have bounded multiplicity. The author suggests that the result might hold without conditions (iii) and (iv). Another result deals with the distance between non-real poles of a meromorphic function \(f\) for which \(f\) and \(f''\) have only finitely many non-real zeros.
scientific article; zbMATH DE number 6049595
scientific article