Dear Omran Kouba

We need to discuss your question step by step.

First, we have |α| < λ ≤ |Q(eit)|/ |P(eit)| = |eimtQ(eit)|/ |P(eit)|

(by using |eimtQ(eit)| = |Q(eit)| )

therefore, |α||P(eit)| < |eimtQ(eit)| .

It is not clear to me why you need (Rouche' theorem, and m is nonnegative)!

Best regards

Dear Issam Kaddoura

m is positive otherwise Rα,m would not be a polynomial. By Rouché’s theorem the inequality reads |Rα,m -XmQ|

Dear Omran Kouba ,

If m is irrational, then Xm is a multiple-valued function.

Rouché’s theorem is applied over holomorphic functions,

how one overcome this condition?

Dear Issam Kaddoura

I assumed m is a nonnegative integer, I am interested in this case only,

in this case i5 cannot be an irrational number.

Dear Omran Kouba

According to the previous construction,

we have m independent of α and - λ < α < λ.

The set ∩m (Rα,m) includes ( - λ, λ ). If we add any value for α outside the interval ( - λ, λ ) the conditions to apply Rouché’s theorem are not satisfied.

So, we need other criteria to find bounds of the roots of Rr,m for fixed m and r is outside the interval ( - λ, λ ), and this new criterion is strongly based on the coefficients of both Q(x) and P(x).

Dear Mercedes Orús-Lacort

Rouché’s theorem gives an implication, not an equivalence. If the inequality is true then Rα,m has as much zeros in the unit disk as we want so if α is in the (-λ,λ) then surely α is in Am but Am may be larger than that, it need not be symmetric, and I do not know if it is an interval containing (-λ,λ) which is what I ask to be proved.

Dear Omran Kouba

I think your claim is valid.

Proof:

R_{α,m} = x^{m}Q(x) - αP(x)

A_{m} = {α∈R, Z(R_{α,m})∈D(0,1)}, Z(R_{α,m}) the set of zeros of R_{α,m}.

Obviously, (-λ,λ) ⊂ A_{m}≠ φ ( by construction).

Let a, b ∈ A_{m}, required to prove that [a,b] ⊂ A_{m}.

a ∈ A_{m} ⇒ x^{m}Q(x) - aP(x) satisfies the boundary condition say B(a),

that gaurantees Z(R_{a,m}) ⊂ D(0,1).

b ∈ A_{m} ⇒ x^{m}Q(x) - bP(x) satisfies the boundary condition say B(b),

that gaurantee Z(R_{b,m}) ⊂ D(0,1).

[ You can use any boundary conditions that show the boundaries of the roots

for example: see

https://en.wikipedia.org › wiki › Geometrical_properties_of_polynomial_r...]

Let a < c < b, obviously if B(a) and B(b) are true, then B(c) is true and then

Z(R_{c,m}) ⊂ D(0,1).

Therefore, A_{m} is an interval.

PS: I think the general approach of Peter Breuer deserves to study, I think one can make a more general conjecture.

Best regards