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

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).


Introduction
Consider the trinomial equation where z is a complex number, n and k are two integers such that k = 1,2,...,n − 1, and α is a real number.The first discussion of the behavior of the roots of trinomial equations was in Fell [1].She presented a description of the trajectories of these roots, called trinomial arcs.These arcs can be expressed in polar coordinates (ρ,θ) by a function ρ(θ) and are continuous in functions of α as α varies between 0 and 1, or between 1 and +∞, or between −∞ and 0. Fell [1] also studied the monotonicity of the function α(θ) and gave a bound for the modulus of roots.However, she did not establish the monotonicity of ρ as a function of θ.Though the descriptive results of Fell [1] give us the information about the form and location of the trinomial arcs, nevertheless, these types of arcs are not suitably designed for further study.In this paper, we will restrict our attention to a family of trinomial arcs, solutions of (1.1) with 0 < α < 1, inside the unit disk D u = {z : |z| ≤ 1}.We begin by defining this family of trinomial curves denoted by B(n,k,r), where n, k, and r satisfy some conditions.Note that Dubuc and Zaoui [2] studied trinomial arcs denoted by B m and which are part of this family of arcs B(n,k,r).Next, we prove in this work that ρ(θ) is a differentiable function for these arcs.With a view to solving the problem of monotonicity of ρ(θ) for the trinomial arcs B(n,k,r), two important intermediate results are shown.At last, this study allows us to prove that ρ(θ) is a decreasing function.

Study of the trinomial equation
In (1.1), fix n and k.For z = ρe iθ in (1.1), one has ρ n e inθ = αρ k e ikθ + (1 − α).Separating real and imaginary parts, one gets ρ n sinnθ = αρ k sinkθ and ρ n cos nθ = αρ k cos kθ + (1 − α).So, when θ / = lπ/n, where l is an integer, we get On the other hand, divide (1.1) by z n and consider the imaginary part.When α / = 0 and θ / = lπ/(n − k), where l is an integer, we obtain that Therefore, we have the next equation of the trajectories of roots of (1.1): In fact, Fell has studied in [1] the trinomial equation where z is a complex number, n and k are two integers such that k = 1,2,...,n − 1, and λ is a real number.Substituting into (2.4) the expression given for z n by (1.1), we get When z is not a kth root of unity, it follows that α = 1 − 1/λ.Hence, in order to pass from (1.1) to (2.4), we can set α = 1 − 1/λ.It stems easily from this equality that the case 0 ≤ α ≤ 1 of (1.1) corresponds to the case 1 ≤ λ < +∞ of (2.4).
Remark 2.3.The upper and lower half-planes are symmetrical.So, we will restrict our study of trinomial arcs to the upper half-plane.

Description and definition of trinomial arcs B(n,k,r)
Notice that for α = 0, (1.1) has n roots: the nth roots of unity.Fell [1], in her Descriptive Claim II, pages 314-315, tells us that the trajectories of the n roots can be described as trajectories of particles starting at these n roots.As α changes from 0 to 1, they move continuously until α = 1, (n − k) of them have moved into (n − k)th roots of unity, and k of them have collapsed to 0. There are k trajectories going to 0, the k tangents being lines going through 0 and one kth root of −1.Consider C = {nth roots of unity}, D = {(n − k)th roots of unity}, and E = {kth roots of − 1}.Let γ be in C and let δ be the unique nearest neighbor of γ in D ∩ E. Fell [1] asserts that, in the case δ ∈ D ∩ E with 0 ≤ α ≤ 1, there exists γ in C such that δ is equidistant from γ and from γ .There exists also α 0 in [0,1] such that the trajectories of two particles starting at γ and γ when α = 0 are continuous arcs until the point of their meeting on the line segment θ = arg(δ) when α = α 0 .When α moves from α 0 to 1, the two roots remain on the segment θ = arg(δ), one of them goes to 0 and the other tends to δ. Fell shows in [1] that all the trinomial arcs solutions of (1.1) in the case 0 ≤ α ≤ 1 with δ ∈ D ∩ E are such that the feasible angles θ belong to intervals of length less than or equal to π/n and bounded on the one side by arg(δ) where δ is both a kth root of −1 and an (n − k)th root of unity, and on the other side by arg(γ) where γ is an nth root of unity.There are so two types of arcs in this case; the first type is such that θ belongs to [arg(γ),arg(δ)] where γ ∈ C and the second type is such that θ belongs to [arg(δ),arg(γ )] where γ ∈ C, such that δ is equidistant from γ and from γ .
In [2], Dubuc and Zaoui studied a class of trinomial arcs denoted by B m and defined as the set of roots of (1.1) with 0 ≤ α ≤ 1, n = m, k = m − 2, where m is an odd integer larger than 2 and the feasible angles belong to the interval [π − π/m,π].They showed in [2] that ρ(θ) is a decreasing function on [π − π/m,π] for the arcs B m .Because m is an odd integer, we can say that γ such that arg(γ) = π − π/m is an nth root of unity and δ such that arg(δ) = π is both a kth root of −1 and an (n − k)th root of unity.Dubuc and Zaoui have thus solved the problem of monotonicity of ρ(θ), pointed out in [1? ], for some particular trinomial arcs, namely B m , solutions of (1.1) in the case 0 ≤ α ≤ 1 with δ ∈ D ∩ E and θ ∈ [arg(γ),arg(δ)].In this paper, our objective is to study the monotonicity of ρ(θ) for all trinomial arcs corresponding to this case.In fact, these arcs, denoted by B(n,k,r), will be defined as follows.
Definition 3.2.Let n and k be two integers such that n is greater than or equal to 3 and 0 < k < n.Let d = gcd(k,n) and r ∈ {0, 1,...,d − 1}, one assumes that k/d and n/d are odd numbers, then the continuous arc is the set of roots of (1.1) with 0 < α < 1.

Differentiability of the function ρ(θ) for the arcs B(n,k,r)
Now, we will prove that the derivative dρ/dθ exists and it is well defined for the trinomial arcs B(n,k,r).

Conclusion
In this work, we have studied the behavior of the family of trinomial arcs B(n,k,r), composed of all solutions of (1.1) in the case 0 < α < 1 with the feasible angles θ in the interval [arg(γ),arg(δ)], where γ is an nth root of unity and δ is both a kth root of −1 and an (n − k)th root of unity.The problem of monotonicity of the trinomial arcs is completely solved in this case.During the description and definition of B(n,k,r), we have evoked another type of trinomial arcs, defined as the solutions of (1.1) in the case 0 < α < 1 with the feasible angles θ in the interval [arg(δ),arg(γ )], where δ is both a kth root of −1 and