On a theorem of Baernstein
\textit{A. Baernstein II} (J. Anal. Math. 53, 253-268 (1989; Zbl 0689.30017) considered the following problem. Let \(\Theta\) be the complement on the sphere of the continuum consisting of the union of the ray on the real axis from \(- \infty\) to 1 and the segment from 0 to \(a = \exp i \pi/3\). Let \(F_1 (z)\), \(F_2 (y)\) map \(\Theta\) onto the plane slit along the negative real axis such that \(F_1 (1) = 0\), \(F_2 (a) = 0\), \(\lim_{z \to \infty} |F_i (z)/z |= 1\), \(i = 1,2\). Let \(y = \lim_{z \to 1} |{F_1 (z) \over z - 1} |\); \(\beta = \lim_{z \to a} |{F_2 (z) \over z - a} |\). He asserts that \(y^{1/2} + \beta^{1/2} > \sqrt 2\). Baernstein's presentation depends on complicated estimates and numerical calculations. The present author gives a conceptual proof using the method of the extremal metric and techniques introduced by \textit{K. Oikawa} and the reviewer [Ill. J. Math. 15, 664-671 (1971; Zbl 0229.30012)].
scientific article; zbMATH DE number 792860
scientific article