An optimization problem and point-evaluation in Paley-Wiener spaces

We study the constant ${\mathcal{C}}_p$ defined as the smallest constant $C$ such that $\mathrm{\mid }f(0){\mathrm{\mid }}^p\le C\mathrm{\parallel }f{\mathrm{\parallel }}_p^p$ holds for every function $f$ in the Paley-Wiener space $P{W}^p$. Brevig, Chirre, Ortega-Cerdà, and Seip have recently shown that ${\mathcal{C}}_p<p\mathrm{/}2$ for all $p>2$. We improve this bound for $2<p\le 5$ by solving an optimization problem.