Let \(G\) be the group of conformal automorphisms of the complex unit disk \(D\) and let \(G=KAN\) be its Iwasawa decomposition. The author finds such a contour \(Q\subset D\) that the condition \[ \int\limits_{\partial (gQ)}f(z) dz=0\quad \text{for all }g\in NA, \] together with some assumptions regarding the asymptotic behaviour of a continuous function \(f\) as \(z\to 1\), implies holomorphy of \(f\).