On a holomorphy criterion for an arbitrary mapping of a planar domain into the plane
The author derives a sufficient condition for a one-to-one mapping \(f\: D\to\mathbb C\) from a domain \(D\) into the complex plane \(\mathbb C\) to be holomorphic. This condition generalizes a condition obtained by \textit{D. Menchoff} [Math. Ann. 109, 101-159 (1933; Zbl 0007.41502)] and is closely related to a condition previously obtained by the author [Sib. Math. J. 32, No.1, 21-28 (1991; Zbl 0739.30001)]. A mapping \(f\: D\to\mathbb C\) satisfies condition \(K^{*}\) at a point \(z\in D\) if there exist three rays \(t_i(z)\), \(i=1,2,3\), emanating from \(z\) such that each open half-plane contains at least one of the rays and \[ \lim_{{\Delta z\to 0}\atop{z+\Delta z\in\bigcup^3_{i=1}t_i(z)}} \biggl|{f(z+\Delta z)-f(z) \over \Delta z}\biggr| \] exists and is finite. A mapping \(f\: D\to\mathbb C\) satisfies condition \(K^{**}\) at a point \(z\in D\) if there exist five rays \(t_i(z)\), \(i=1,2,3,4,5\), emanating from \(z\) such that each open half-plane contains at least two of the rays and \[ \lim_{{\Delta z\to 0}\atop{z+\Delta z\in\bigcup^5_{i=1}t_i(z)}}\text{arg} {f(z+\Delta z)-f(z) \over \Delta z} \] exists. The following theorem is proven: Let \(f\: D\to\mathbb C\) be a bounded one-to-one mapping which satisfies either condition \(K^{*}\) or condition \(K^{**}\) at each point \(z\in D\). Then either \(f\) or \(\overline f\) is holomorphic in \(D\).
scientific article; zbMATH DE number 1099566