Polynomial Inequalities, Mahler’s Measure, and Multipliers

We survey polynomial inequalities obtained via coefficient multipliers, for norms defined by the contour or the area integrals over the unit disk. Special attention is devoted to the Szegő composition and the inequalities related to Mahler’s measure. We also consider a new height on polynomial spaces defined by the integral over the normalized area measure on the unit disk. This natural analog of Mahler’s measure inherits many nice properties such as the multiplicative one. However, this height is a lower bound for Mahler’s measure, and it fails an analog of Lehmer’s conjecture. 1. The Szegő composition and polynomial inequalities This paper is a survey of results on polynomial inequalities obtained via coefficient multipliers, and other topics related to Mahler’s measure. Let Cn[z] and Zn[z] be the sets of all polynomials of degree at most n with complex and integer coefficients respectively. Mahler’s measure of a polynomial Pn ∈ Cn[z] is defined by M(Pn) := exp ( 1 2π ∫ 2π 0 log |Pn(e)| dθ ) . It is also known as the contour geometric mean or as the H Hardy space norm. The latter name is explained by the following relation to the Hardy spaces. Defining the Hardy space norm by ‖Pn‖Hp := ( 1 2π ∫ 2π 0 |Pn(e)| dθ )1/p , 0 < p < ∞, we note [18] that M(Pn) = limp→0+ ‖Pn‖Hp . An application of Jensen’s inequality immediately gives that M(Pn) = |an| ∏ |zj |>1 |zj| 2000 Mathematics Subject Classification. Primary 11C08; Secondary 11G50, 30C10.