Jensens functional and polynomials in several variables
In this paper we obtain some lower bounds of the integral ? 0 2 ? ? ? 0 2 ? log | P ( e i ? 1 ,  , e i ? n ) | d ? 1 2 ? ? d ? n 2 ? if the polynomial P(z1,,zn) P ( z 1 ,  , z n ) in several variables has a concentration at low degrees, measured by: (1) lp l p -norm under lp l p -norm, 1?p?2 1 ? p ? 2 ; (2) lp l p -norm under Lp L p -norm, p>2 p > 2 ; (3) lp l p -quasi-norm under lp l p -quasi-norm, 0<p<1 0 < p < 1 . Presented lower bounds generalized and extended the results from [B. Beauzamy, Jensens inequality for polynomials with concentration at low degrees, Numer. Math. 49 (1986), 221225; M. Pavlovi?, N. Caki?, M. Rajovi?, S. Radenovi?, A generalization of Jensens inequality for polynomials having concentration at low degrees, Comput. Math. Appl. 57 (2009) 332337; and S. Radenovi?, Some estimates of integral ? 0 2 ? log | p ( e i ? ) | d ? 2 ? , Publ. Inst. Math. (Beograd) (NS) 52 (66) (1992) 3742] for polynomials in one variable.
jensens functional polynomial several variable lower bound concentration at low degree