Polynomial inequalities on the $\pi /4$-circle sector

Abstract

A number of sharp inequalities are proved for the space $\mathcal{P}\left({}^{2}D\left(\frac{\pi }{4}\right)\right)$ of 2-homogeneous polynomials on ${\mathbb{R}}^2$ endowed with the supremum norm on the sector $D\left(\frac{\pi }{4}\right):=\left\{e^{i\theta }:\theta \in \left[0,\frac{\pi }{4}\right]\right\}$. Among the main results we can find sharp Bernstein and Markov inequalities and the calculation of the polarization constant and the unconditional constant of the canonical basis of the space $\mathcal{P}\left({}^{2}D\left(\frac{\pi }{4}\right)\right)$.

References in corpus (1)

• The Bohr radius of the $n$-dimensional polydisk is equivalent to $\sqrt{\frac{\log n}{n}}$