## Abstract

Let f be a meromorphic non-entire function in the plane, and suppose that for every k ≥ 0, the derivative f^{(k)} has only real zeros. We have proved that then f(az + b) = P(z)/Q(z) for some real numbers a and b where a ≠ 0, where Q(Z) = z^{n} or Q(z) = (z^{2} + 1)^{n}, n is a positive integer, and P is a polynomial with only real zeros such that deg P ≤ deg Q + 1 ; or f(az + b) = C(z -i)^{-n} or f(az + b) = C(z - α)/(z -i) where a is real and C is a non-zero complex constant. In this paper we provide part of the proof of this theorem, by obtaining the following result. Let f be given by f(z) = g(z)/(z^{2} + 1)^{n} where g is a real entire function of finite order with g(i)g(-i) ≠ 0 and n is a positive integer. If f, f′, and f″ have only real zeros then g is a polynomial of degree at most 2n + 1. Conversely, if f is of this form where p is a polynomial of degree at most 2n with only real zeros, then f^{(k)} has only real zeros for all k ≥ 0. If the degree of g is 2n + 1 then f^{(k)} has only real zeros for all k ≥ 0 if, and only if, f and f′ have only real zeros.

Original language | English (US) |
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Pages (from-to) | 317-388 |

Number of pages | 72 |

Journal | Annales Academiae Scientiarum Fennicae Mathematica |

Volume | 23 |

Issue number | 2 |

State | Published - 1998 |

## ASJC Scopus subject areas

- General Mathematics