Hörmander's Inequality and Point Evaluations in de Branges Space

Let $f$ be an entire function of finite exponential type less than or equal to $\sigma$ which is bounded by $1$ on the real axis and satisfies $f(0)=1$. Under these assumptions Hörmander showed that $f$ cannot decay faster than $\cos (\sigma x)$ on the interval $(-\pi \mathrm{/}\sigma ,\pi \mathrm{/}\sigma )$. We extend this result to the setting of de Branges spaces with cosine replaced by the real part of the associated Hermite-Biehler function. We apply this result to study the point evaluation functional and associated extremal functions in de Branges spaces (equivalently in model spaces generated by meromorphic inner functions) generalizing some recent results of Brevig, Chirre, Ortega-Cerdà, and Seip.