Inequalities on the Mahler measure of a polynomial
The author proves the following inequalities for the Mahler measure \(M(P)\) which can be used to improve results of Bertin and Schinzel: Let \(P(x)\) be a monic polynomial with real coefficients, with all roots real and positive and with \(| P(1)| \geq 1\); then \[ M(P)^{1/d} \geq (1 + (4| P(0)| ^{1/d} + 1)^{1/2})/2. \] If instead, all roots are real and nonzero and \(| P(\pm 1)| \geq 1\) then \[ M(P)^{2/d} \geq (1 + (4| P(0)| ^{2/d} + 1)^{1/2})/2. \] The proofs use the auxiliary function \(f(x) = \log^+(x) - (c\log| x| + (1 - 2c)\log| x-1| )\). Some related inequalities established by similar techniques may be found in the author's article [\textit{V. Flammang}, Can. Math. Bull. 38, 438-444 (1995; Zbl 0842.26011)].
scientific article; zbMATH DE number 1132888