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

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

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).