LOWER BOUNDS FOR MAHLER-TYPE MEASURES OF POLYNOMIALS

. In this note we consider various generalizations of the classical Mahler measure M(P) = exp 0 log |P(ei theta) d theta for complex polynomials P, and prove sharp lower bounds for them. For example, we show that for any monic polynomial P is an element of C[z] satisfying |P (0)| = 1 the quantity M0 (P) = exp 0 max(0, log |P(ei theta)|) d theta is greater than or equal to M(1 + z1 + z2) = 1.381356.. .. This inequality is best possible, with equality being attained for all P(z) = zn + c with n is an element of N and |c| = 1.