F. Wiener's trick and an extremal problem for $H^p$

For $0<p\le \mathrm{\infty }$, let $H^p$ denote the classical Hardy space of the unit disc. We consider the extremal problem of maximizing the modulus of the $k$th Taylor coefficient of a function $f\in H^p$ which satisfies $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^p}\le 1$ and $f(0)=t$ for some $0\le t\le 1$. In particular, we provide a complete solution to this problem for $k=1$ and $0<p<1$. We also study F. Wiener’s trick, which plays a crucial role in various coefficient-related extremal problems for Hardy spaces.