Behavior of the trinomial arcs \(B(n,k,r)\) when \(0<\alpha <1\) (Q925476)

Summary: We deal with the family \(B(n,k,r)\) of trinomial arcs defined as the set of roots of the trinomial equation \(z^n=\alpha z^k+(1-\alpha)\), where \(z=\rho e^{i\theta}\) is a complex number, \(n\) and \(k\) are two integers such that \(0<k<n\), and \(\alpha\) 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 \((\rho,\beta)\). The question is to prove that \(\rho(\theta)\) is a decreasing function, for each trinomial arc \(B(n,k,r)\).