Strips and hyperbolas for zeros of polynomials in terms of their Hermite expansion. (English) Zbl 1208.12001
In 1950 P. Turán used the expansion of complex polynomials in terms of Hermite polynomials \(H_n\) to study the location of their zeros in certain strips.
Here, among other results, the authors prove the following Theorem. Assume that a complex polynomial \(f(x)\) has the Hermite expansion \(f(x)=\sum_0^n b_jH_j(x)\) and let \(\mu_1\), …, \(\mu_n\) be positive real numbers such that \(\mu_1+ \cdots+\mu_n\leq1\). Then all the roots of \(f\) lie in the strip \[ |\operatorname{Im} z|\leq {1\over2}\,\max_{1\leq j\leq n}\left( |b_{n_j}|\over \mu_j |b_n| \right)^{1/j}. \] The proofs of these new results combine some ideas of Turán and the classical results of Fujiwara, Bailleu, Cowling and Thron.
Here, among other results, the authors prove the following Theorem. Assume that a complex polynomial \(f(x)\) has the Hermite expansion \(f(x)=\sum_0^n b_jH_j(x)\) and let \(\mu_1\), …, \(\mu_n\) be positive real numbers such that \(\mu_1+ \cdots+\mu_n\leq1\). Then all the roots of \(f\) lie in the strip \[ |\operatorname{Im} z|\leq {1\over2}\,\max_{1\leq j\leq n}\left( |b_{n_j}|\over \mu_j |b_n| \right)^{1/j}. \] The proofs of these new results combine some ideas of Turán and the classical results of Fujiwara, Bailleu, Cowling and Thron.
Reviewer: Maurice Mignotte (Strasbourg)
MSC:
12D10 | Polynomials in real and complex fields: location of zeros (algebraic theorems) |
26C10 | Real polynomials: location of zeros |
11R09 | Polynomials (irreducibility, etc.) |
30C10 | Polynomials and rational functions of one complex variable |