Generalizations of Gonçalves' inequality

Let $F(z)=\sum_{n=0}^N a_n z^n$ be a polynomial with complex coefficients and roots $\alpha_1$, ..., $\alpha_N$, let $\Vert F\Vert _p$ denote its $L_p$ norm over the unit circle, and let $\Vert F\Vert _p$ denote Mahler's measure of $F$. Gonçalves' inequality asserts that

$\Vert F\Vert _2$	$\geq \vert a_N\vert \left( \prod_{n=1}^N \max\{1, \vert\alpha_n\vert^2\} + \prod_{n=1}^N \min\{1, \vert\alpha_n\vert^2\} \right)^{1/2}$
	$= \Vert F\Vert _0\left(1+\frac{\vert a_0 a_N\vert^2}{\Vert F\Vert^4}\right)^{1/2}.$

We prove that

$\Vert F\Vert _p \geq B_p \vert a_N\vert \left( \prod_{n=1}^N \m... ...lpha_n\vert^p\} + \prod_{n=1}^N \min\{1, \vert\alpha_n\vert^p\} \right)^{1/p}$

for $1\leq p\leq2$, where $B_p$ is an explicit constant, and that

$\Vert F\Vert _p \geq \Vert F\Vert _0 \left(1+\frac{p^2\vert a_0 a_N\vert^2}{4\Vert F\Vert^4}\right)^{1/p}$

for $p\geq1$. We also establish additional lower bounds on the $L_p$ norms of a polynomial in terms of its coefficients.