Generalizations of Goncalves' inequality

If $F$ is a polynomial with complex coefficients, leading term $a_N$, and roots $\alpha_1$, ..., $\alpha_N$, then Gon\c{c}alves' inequality states that $\|F\|_2^2$ is bounded below by $\abs{a_N}^2 (\prod_{n=1}^N \max\{1, \abs{\alpha_n}^2\} + \prod_{n=1}^N \min\{1, \abs{\alpha_n}^2\})$. We establish generalizations of this inequality for other $L_p$ norms, and derive additional lower bounds on the $L_p$ norms of a polynomial in terms of its coefficients.