Best approximations for the weighted combination of the Cauchy--Szegö kernel and its derivative in the mean

In this paper, we study an extremal problem concerning best approximation in the Hardy space $H^1$ on the unit disk $\mathbb{D}$. Specifically, we consider weighted combinations of the Cauchy-Szegö kernel and its derivative, parametrized by an inner function $\phi$ and a complex number $\lambda$, and provide explicit formula of the best approximation $e_{\phi ,z}(\lambda )$ by the subspace $H_0^1$. We also describe the extremal functions associated with this approximation.