A number of sharp inequalities are proved for the space ${mathcal P}left(^2Dleft(frac{pi}{4}right)right)$ of 2-homogeneous polynomials on ${mathbb R}^2$ endowed with the supremum norm on the sector $Dleft(frac{pi}{4}right):=left{e^{itheta}:thetain le
ft[0,frac{pi}{4}right]right}$. 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 ${mathcal P}left(^2Dleft(frac{pi}{4}right)right)$.
In this paper we prove that the complex polynomial Bohnenblust-Hille constant for $2$-homogeneous polynomials in ${mathbb C}^2$ is exactly $sqrt[4]{frac{3}{2}}$. We also give the exact value of the real polynomial Bohnenblust-Hille constant for $2$-h
omogeneous polynomials in ${mathbb R}^2$. Finally, we provide lower estimates for the real polynomial Bohnenblust-Hille constant for polynomials in ${mathbb R}^2$ of higher degrees.
