Some isoperimetric inequalities and their application to problems on polynomials

Abstract

A b s t r a c t . We prove sharp inequalities for the product of the Lebesgue integral of certain power of a positive absolutely continuous and nondecreasing function (or a linear combination of such distinct powers) and the Lebesgue integral of the square of its derivative. These inequalities are related to some problems for polynomials having small Mahler measure. As an application, we give a lower bound for the logarithmic height of a noncyclotomic algebraic number in terms of its degree.