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

A number of sharp inequalities are proved for the space ${\mathcal P}\left(^2D\left(\frac{\pi}{4}\right)\right)$of2-homogeneouspolynomialson${\mathbb R}^2$endowedwiththesupremumnormonthesector$D\left(\frac{\pi}{4}\right):=\left{e^{i\theta}:\theta\in \left[0,\frac{\pi}{4}\right]\right}$. 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 $\mathcal{P}\left({}^{2}D\left(\frac{\pi }{4}\right)\right)$.