  In this paper we address the following question: given a holomorphic function
with prescribed $L^p(\mathbb{R})$ and $L^q(\mathbb{R})$ norm (with $1\leq p,q
\leq \infty$) along two parallel lines in the complex plane, then what is the
maximum value that this function can achieve at a given point between these
lines. Here we show that this problem is well-posed in suitable Hardy-like
spaces on the strip. Moreover, in this setting we completely solve this problem
by providing not only an explicit formula for the optimizers but also for the
optimal values. In addition, we briefly discuss some applications of these
results to interpolation theory and to Lieb-Thirring inequalities.
