Generalizations of Goncalves' inequality

Abstract

If $F$ is a polynomial with complex coefficients, leading term $a_N$, and roots ${\alpha }_1$, ..., ${\alpha }_N$, then Gonçalves' inequality states that $\Vert F{\Vert }_2^2$ is bounded below by $\abs{a_N}^2 (\prod_{n=1}^N \max\{1, \abs{α_n}^2\} + \prod_{n=1}^N \min\{1, \abs{α_n}^2\})$. 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.

9 pages