On a criterion of conformality of a mapping of simply connected domains (Q5942132)

Let \(D\) and \(G\) be two simply connected domains with smooth boundaries \(\Gamma\) and \(L\), respectively. If \(\varphi\) is an analytic function in \(D\), continuous in the closure \(\overline{D}\) mapping \(\Gamma\) in one-to-one way onto \(L\) preserving orientation, then it is well known that \(\varphi\) maps \(D\) conformally onto \(G\). The main result of this article is a new criterion for conformality whose conditions are easier to verify. The result reads as follows. Let \(D\) and \(G\) be as above, and let \(\varphi\) be an analytic function in \(D\) which is continuously differentiable in \(\overline{D}\). Furthermore, let \(z_0 \in D\), \(\zeta_0 \in G\), \(z_1 \in \Gamma\) and \(\zeta_1 \in L\) such that \(\varphi\) satisfies \(\varphi(z) \in L\) for \(z \in \Gamma\), \(\varphi(z_0)=\zeta_0\), \(\varphi(z_1)=\zeta_1\), \(\varphi'(z_1) \neq 0\) and \(\varphi(z) \neq \zeta_1\) for \(z \in \Gamma\) with \(z \neq z_1\). Then \(\varphi\) maps \(D\) conformally onto \(G\).