Estimates on the norm of polynomials and applications

In this paper, equivalence constants between various polynomial norms are calculated. As an application, we also obtain sharp values of the Hardy–Littlewood constants for $2$-homogeneous polynomials on ${\mathrm{\ell }}_p^2$ spaces, $2<p\le \mathrm{\infty }$ and lower estimates for polynomials of higher degrees.