Lower bounds for Mahler measures of certain special polynomials (Q5939700)

Let \(P(x) = ax^d + \dots \pm b\) be a polynomial of degree \(d \geq 1\) with real coefficients and with \(a, b > 0\). Let \(p = |P(1)|\), \(q = |P(-1)|\) and \(r = |P(i)|\), and let \(M\) denote the Mahler measure of \(P\). The author proves in a very simple way a number of interesting inequalities, for example that if all of the roots of \(P\) are positive and if \(p > 0\) then \(M^{1/d} \geq (p^{1/d} + q^{1/d})/2\), and if all roots are real and \(p,q,r > 0\) then \(M^{2/d} \geq ((pq)^{1/d} + r^{2/d})/2\). He shows that these are stronger than the related results of \textit{V. Flammang} [J. Théor. Nombres Bordx. 9, 69--74 (1997; Zbl 0892.11035)]. The main ingredient in the proof is the following simple consequence of the arithmetic-geometric mean inequality: if \(a_i, b_i > 0\) for \(i = 1,\dots,n\) then NEWLINE\[NEWLINE(a_1\cdots a_n)^{1/n} + (b_1 \cdots b_n)^{1/n} \leq ((a_1+b_1)\cdots(a_n+b_n))^{1/n}.NEWLINE\]