Analytic theory of polynomials

TL;DR: In this paper, a sharp inequality for trigonometric polynomials with a double zero at a specified point was obtained, and a similar extremal problem has been studied by R.P. Boas.

TL;DR: The subject of extremal problems for polynomials and related classes of functions plays an important and crucial role in obtaining inverse theorems in approximation theory as discussed by the authors, and the further progress in inverse theorem extraction has depended upon first obtaining the corresponding analogue or generalization of Markov's and Bernstein's inequalities, and these inequalities have been the starting point of a considerable literature in Mathematics.

TL;DR: This work unifies several well-known matrix problems and to consider common methods of their analysis in one concept of $({\mathfrak D}, {\mathcal G}, \circ)$-stability, which depends on a stability region ${\math frak D} \subset {\mathbb C}$, a matrix class ${\ mathcal G}$ and a binary matrix operation $\circ$.