Let \({\mathfrak L}_{\Gamma}\) denote the set of conformal mappings \(\phi\) of the disc \(E=\{z:| z| <1\}\) into itself which extend analytically to the arc \(\Gamma =\{z=e^{i\theta}:\) \(-\frac{\pi}{2}<\theta <\frac{\pi}{2}\}\) and satisfy the condition \(\phi (\Gamma)=\Gamma\). Since a composition of mappings from the set \({\mathfrak L}_{\Gamma}\) belongs to \({\mathfrak L}_{\Gamma}\), therefore \({\mathfrak L}_{\Gamma}\) is a semigroup. Let f be a conformal mapping of the strip \(\Pi =\{z:\) \(0