International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 91535, 8 pages
doi:10.1155/2007/91535
Research Article

Behavior of the Trinomial Arcs B(n,k,r) when 0<α<1

Kaoutar Lamrini Uahabi1 and Mohammed Zaoui2

1Forces Armées Royales Boulevard 49, Apartment no. 9, Nador 62000, Morocco
2Department of Mathematics, Faculty of Sciences, Mohamed first University, P.O. Box 524, Oujda 60000, Morocco

Received 12 April 2007; Revised 14 July 2007; Accepted 12 August 2007

Academic Editor: Teodor Bulboaca

Copyright © 2007 Kaoutar Lamrini Uahabi and Mohammed Zaoui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We deal with the family B(n,k,r) of trinomial arcs defined as the set of roots of the trinomial equation zn=αzk+(1−α), where z=ρeiθ is a complex number, n and k are two integers such that 0<k<n, and α is a real number between 0 and 1. These arcs B(n,k,r) are continuous arcs inside the unit disk, expressed in polar coordinates (ρ,θ). The question is to prove that ρ(θ) is a decreasing function, for each trinomial arc B(n,k,r).