2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73819If $F$ is a polynomial with complex coefficients, leading term $a_N$, and roots $α_1$, ..., $α_N$, then Gonçalves' inequality states that $\|F\|_2^2$ 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 pagesClassical Analysis and ODEsNumber Theory30A10, 30C10 (Primary) 26D05, 42A05 (Secondary)text