Numer. Math. 49, 221-225 (1986) Numerische Mathematik 9 Springer-Verlag 1986 Jensen's Inequality for Polynomials with Concentration at Low Degrees Bernard Beauzamy Department de Math6matiques, Universit6 Claude Bernard-Lyon i, 43, Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France Summary. We give a generalization of Jensen's Inequality, valid for poly- nomials having some concentration at low degrees. We investigate the constants involved, both from a theoretical and a numerical point of view. Subject Classifications: AMS(MOS): G5D99; CR: G1.2. Let P be a polynomial with complex coefficients: P(x) = a o + a 1 x + a 2 X 2 -~ a 3 x 3 + . . . (1) and let d, 0 < d < 1. We say that P has concentration d at degrees at most k if j=k la~l >d ~ I@. (2) j=o j>_-o In the sequel, we shall normalize P and assume that Z laj I =1. (3) j_->o The aim of this paper is to give an extension of the classical Jensen's In- equality, valid for polynomials satisfying (2) and (3); we shall also give some numerical estimates for the constants involved. Theorem 1. For all k>0, all d, 0<d< 1, all polynomials satisfying (2) and (3), we have: 2~ dO> Log IP(ei~ C(d, k), o 2~= where C(d, k) is the maximum value of the following function: 2d fa'k(t)=t L~ (t--l) \ \~-- l [t + l ]k +' -- l ) (4)