Estimates of the norm of the holomorphic components of functions meromorphic in domains with a smooth boundary

Let $D$ be a Jordan domain with smooth boundary, $f$ a function meromorphic in $D$, and $f^*$ the holomorphic component of $f$ in $D$. It is shown that $\|f^*\|_{\partial D}\leqslant C(D)n\|f\|_{\partial D}$, where $n$ is the number of poles of $f$ in $D$.
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