Generalizations of Gonçalves’ inequality

Abstract:

Let $F(z)=\sum _{n=0}^N a_n z^n$ be a polynomial with complex coefficients and roots $\alpha _1$, …, $\alpha _N$, let $\|F\|_p$ denote its $L_p$ norm over the unit circle, and let $\|F\|_p$ denote Mahler’s measure of $F$. Gonçalves’ inequality asserts that \begin{align*} \|F\|_2 &\geq |a_N| \left ( \prod _{n=1}^N \max \{1, |\alpha _n|^2\} + \prod _{n=1}^N \min \{1, |\alpha _n|^2\} \right )^{1/2} &= \|F\|_0\left (1+\frac {|a_0 a_N|^2}{\|F\|^4}\right )^{1/2}. \end{align*} We prove that \[ \|F\|_p \geq B_p |a_N| \left ( \prod _{n=1}^N \max \{1, |\alpha _n|^p\} + \prod _{n=1}^N \min \{1, |\alpha _n|^p\} \right )^{1/p} \] for $1\leq p\leq 2$, where $B_p$ is an explicit constant, and that \[ \|F\|_p \geq \|F\|_0 \left (1+\frac {p^2|a_0 a_N|^2}{4\|F\|^4}\right )^{1/p} \] for $p\geq 1$. We also establish additional lower bounds on the $L_p$ norms of a polynomial in terms of its coefficients.