On the analytic functions

Let $\mathrm{\Omega }$ be a connected bounded domain on the complex plane, $S$ be its boundary, which is closed, star-shaped, $C^1$-smooth, and $H(\mathrm{\Omega })$ is the set of analytic (holomorphic) in $\mathrm{\Omega }$ functions. The aim of this paper is to prove that an arbitrary $f\in L^1(S)$, satisfying the condition ${\int }_{S}f(s)ds=0$, can be boundary value of an $f\in H(\mathrm{\Omega })$.