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Abstract. Let Pn be the complex vector space of all polynomials of degree at most n. We give several characterizations of the linear operators T : Pn → Pn for which there exists a constant C > 0 such that for all nonconstant f ∈ Pn there exist a root u of f and a root v of Tf with |u−v| ≤ C. We prove that such perturbations leave the degree unchanged and,… (More)

- VANIA MASCIONI
- 2008

We provide a unified, elementary, topological approach to the classical results stating the continuity of the complex roots of a polynomial with respect to its coefficients, and the continuity of the coefficients with respect to the roots. In fact, endowing the space of monic polynomials of a fixed degree n and the space of n roots with suitable topologies,… (More)

- VANIA MASCIONI
- 2008

Given the harmonic mean μ of the numbers xi (i = 1, 2, 3) and a t ∈ (0,min{x1, x2, x3}/μ}), we determine the best power mean exponents p and q such that Mp(xi − tμ) ≤ (1 − t)μ ≤ Mq(xi − tμ), where p and q only depend on t. Also, for t > 0 we similarly handle the estimates Mp(xi + tμ) ≤ (1 + t)μ ≤Mq(xi + tμ).

- Vania Mascioni
- Electr. J. Comb.
- 2004

In the context of finite metric spaces with integer distances, we investigate the new Ramsey-type question of how many points can a space contain and yet be free of equilateral triangles. In particular, for finite metric spaces with distances in the set {1, . . . , n}, the number Dn is defined as the least number of points the space must contain in order to… (More)

Given a polynomial p of degree n ≥ 2 and with at least two distinct roots let Z(p) = {z : p(z) = 0}. For a fixed root α ∈ Z(p) we define the quantities ω(p, α) := min { |α − v| : v ∈ Z(p) \ {α} } and τ(p, α) := min { |α − v| : v ∈ Z(p′) \ {α} } . We also define ω(p) and τ(p) to be the corresponding minima of ω(p, α) and τ(p, α) as α runs over Z(p). Our main… (More)

- VANIA MASCIONI
- 2012

We prove a tight inequality of non-exponential type for a weighted geometric mean commonly appearing when using Stirling’s approximation (also frequently studied in its logarithmic form when computing entropies). As an application we prove corollaries involving binomial coefficients. 1. The main inequalities If r ∈ R\ {0} , in the notation from Mitrinović’s… (More)

The main result here is a simple general-purpose numerical inequality that can be used to produce a variety of Aczél type inequalities with little effort.

Consider the real vector space P2 of all polynomials of degree at most 2. High-school students study the roots of the polynomials in P2, while linear algebra students study linear transformations on P2. Is it possible to bring these two groups together to do some joint research? For example, a linear algebra student chooses a specific linear transformation… (More)

- VANIA MASCIONI
- 2008

Let T = α0I + α1D + · · · + αnD n , where D is the differentiation operator and α0 = 0, and let f be a square-free polynomial with large minimum root separation. We prove that the roots of T f are close to the roots of f translated by −α1/α0.

The celebrated Gauss-Lucas theorem states that all the roots of the derivative of a complex non-constant polynomial p lie in the convex hull of the roots of p, called the Lucas polygon of p. We improve the Gauss-Lucas theorem by proving that all the nontrivial roots of p′ lie in a smaller convex polygon which is obtained by a strict contraction of the Lucas… (More)