On Mahler measures of a self-inversive polynomial and its derivative
Abstract
Let M ( f ) be the Mahler measure of a polynomial f ∈ ℂ ( z ). An old inequality (which is due to Mahler himself) asserts that M ( f ′) ≤ dM ( f ) for each f ∈ ℂ ( z ) of degree d . In contrast, we prove that if f ∈ ℂ ( z ) is a self-inversive polynomial of degree d ≥ 2, then M ( f ′)>( d /2) M ( f ). We also show that this inequality is best possible for d even, namely, that the quotient M ( f ′)/ M ( f ) takes every value in the interval ( d /2, d ) as f runs through reciprocal polynomials f ∈ ℝ ( z ) of degree d . It seems likely that for d odd the constant d /2 is not optimal. For instance, for d = 3, the optimal value of the constant is 1.93867997… instead of 3/2. For each odd d ≥ 5, we prove that there exists a monic reciprocal polynomial f ∈ ℤ ( z ) of degree d such that M ( f ′)<(( d +1)/2) M ( f ). A corresponding problem for L p -norms of a self-inversive polynomial and its derivative is also considered.