IJPAM: Volume 38, No. 2 (2007)

ON THE MONOTONICITY OF THE TRINOMIAL ARCS
OUTSIDE THE UNIT DISK, WHEN $\alpha >1$

Kaoutar Lamrini Uahabi

, Mohammed Zaoui

F.A.R. Blvd., 49, Apartment No. 9
Nador, 62000, MOROCCO
e-mail: lamrinika@yahoo.fr

Department of Mathematics
Faculty of Sciences
Mohamed First University
P.O. Box 524, Oujda, 60000, MOROCCO
e-mail: zaouimoh@menara.ma

Abstract.In this work, we deal with the family of trinomial arcs defined as the set of roots of trinomial equation $z^n=\alpha z^k+(1-\alpha )$ , where $z=\rho ~e^{i\theta }$ , and are two integers such that $1\leq k\leq n-1$ , $\alpha$ is a real number greater than and and are nonzero integers satisfying some conditions. These curves are continuous arcs expressed in polar coordinates $(\rho,\theta )$ by a function $\rho \left( \theta \right)$ , where $\rho \geq 1$ and $\theta$ is a feasible angle, an angle verifying some conditions. In this paper, the question is to prove that $\rho$ changes monotonically with respect to $\theta$ and that $\rho \left( \theta \right)$ is an increasing function, for each trinomial arc .