Tangent variation principle in complex analysis (Q1804164)

I shall only state the simplest case of the ``Tangent Variation Principle'', omitting the many generalizations given in the paper. Let \(w(z)\) be meromorphic in the closure of a domain \(D \subset \mathbb{C}\) whose boundary in a smooth curve of length \(\ell\). Let \(\Gamma \) be a smooth Jordan arc in the \(w\)-plane and denote by \(L(D, \Gamma)\) the length of \(w^{-1} (\Gamma) \cap D\). Let \(\alpha (P)\) be the angle between the real axis and the tangent to \(\Gamma\) at \(P\) (defined with obvious continuity conditions) and let \(v(\Gamma)\) be the variation of \(\alpha (P)\) as \(P\) describes \(\Gamma\) once. Write \(v_ x (D) [v_ y(D)]\) for \(v (\Gamma_ x) [v(\Gamma_ y)]\), where \(\Gamma_ x\) is the intersection of \(\text{Re} w = x[\text{Im} w = y]\) with \(D\). Then \[ L(D, \Gamma) \leq K (\Gamma) \Bigl( \int v_ x (D)dx + \int v_ y (D) dy + \ell \Bigr). \] Many consequences of this ``Tangent Variation Principle'' are given. The work has obvious affinities to the work done by Ahlfors, Beurling and many others on extremal length and related topics. The author gives no references to texts or papers on these subjects; he may not be aware of them. It seems likely to the reviewer that the Tangent Variation Principle and more can be deduced from the theory of extremal length.