Jensen’s functional and polynomials in several variables

In this paper we obtain some lower bounds of the integral ∫02π⋯∫02πlog|P(eiθ1,…,eiθn)|dθ12π⋯dθn2π if the polynomial P(z1,…,zn)P(z1,…,zn) in several variables has a concentration at low degrees, measured by: (1) lplp-norm under lplp-norm, 1≤p≤21≤p≤2; (2) lplp-norm under LpLp-norm, p>2p>2; (3) lplp-quasi-norm under lplp-quasi-norm, 0