Generalizations of Goncalves' inequality

Borwein, Peter; Mossinghoff, Michael J.; Vaaler, Jeffrey D.

Generalizations of Goncalves' inequality

Authors

Borwein, Peter

Mossinghoff, Michael J.

Vaaler, Jeffrey D.

Description

If $F$ 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 pages