THE RAMANUJAN JOURNAL, Vol. 14, No. 2 (2007), 305-320.

Self-inversive polynomials of odd degree

LÁSZLÓ LOSONCZI
Faculty of Economics and Business Administration, University of Debrecen, 4028 Debrecen, Kassai u. 26, Hungary
Email: losi@math.klte.hu
Homepage: http://riesz.math.klte.hu/~losi/

ANDRZEJ SCHINZEL
Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
Email: A.Schinzel@impan.gov.pl

Received January 13, 2004; Accepted February 22, 2006

Abstract. If the coefficients of a self-inversive polynomial $P(z) = \sum_{k=0}^m A_k z^k \in \C[z]$ of odd degree $m \ge 3$ satisfy the inequality
$$

\abs{A_m} \ge \cos \frac{\pi}{2(m + 1)} \mbox{inf}_{c,d\in\C, \abs{d}=1} \sum_{k=0}^m \abs{c A_k - d^{m-k} A_m} $$
then all zeros of $P$ are on the unit circle and they are simple.

This is an improvement of a recent result of the second author (Ramanujan J. 9 , 19-23, 2005) on the zeros of self-inversive polynomials in the case of polynomials of odd degree. A similar improvement in the case of real (reciprocal) polynomials has been given by Lakatos and the first author (J. Inequal. Pure Appl. Math. 4 (3), 2003).

Keywords: Self-inversive polynomials, zeros

2000 Mathematics Classification: Primary 30C15; Secondary 26C15

Original article available at www.springerlink.com:
http://springerlink.metapress.com/openurl.asp?genre=article&id=doi:10.1007/s11139-007-9029-5

Preliminary Version.
Updated Sat Sep 22 15:09:50 EDT 2007.