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Polynomial inequalities on the $π/4$-circle sector

A number of sharp inequalities are proved for the space ${\mathcal P}\left(^2D\left(\fracπ{4}\right)\right)$ of 2-homogeneous polynomials on ${\mathbb R}^2$ endowed with the supremum norm on the sector $D\left(\fracπ{4}\right):=\left\{e^{iθ}:θ\in \left[0,\fracπ{4}\right]\right\}$. 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 ${\mathcal P}\left(^2D\left(\fracπ{4}\right)\right)$.