The Kraus inequality for multivalent functions (Q679872)

The Schwarzian derivative at an arbitrary point \(z\) in the unit disc is defined for an analytic function \(f\), with non-zero \(f'(z)\), as \[ Sf(z) = \left(\frac{f''(z)}{f'(z)}\right)' - \frac{1}{2} \left(\frac{f''(z)}{f'(z)}\right)^2. \] Well over eight decades ago, \textit{W. Kraus} [Mitt. Math. Semin. Gießen 21, 1--29 (1932; JFM 58.1142.02)] proved that \(|Sf(0)|\) does not exceed 6, provided \(f\) is univalent in the unit disc. In the paper under review, the author replaces the assumption of univalence with a constraint on the image of the unit disc under \(f\). The work depends heavily on the study of Riemann surfaces [\textit{A. Hurwitz}, Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen. Berlin-Göttingen-Heidelberg-New York: Springer-Verlag (1964; Zbl 0135.12101)] and extremal decompositions [\textit{G. V. Kuz'mina}, St. Petersbg. Math. J. 9, No. 3, 1 (1997); translation from Algebra Anal. 9, No. 3, 41--103 (1997; Zbl 0917.30001); St. Petersbg. Math. J. 9, No. 5, 1 (1997); translation from Algebra Anal. 9, No. 5, 1--50 (1997; Zbl 0917.30002)], as well as the author's clever extension of the Lavrentiev inequality for the inner radii of two disjoint domains on the Riemann sphere. With the parameter \[ k = -\frac{1}{2} \arg\frac{Sf(0)}{(f'(0))^2}, \] the main result of the paper follows from the inequality \[ \mathrm{Re}\left[e^{2ki}\frac{Sf(0)}{(f'(0))^2}\right] \leq \frac{6}{|f'(0)|^2}, \] proven in the paper for any real number \(k\) and an analytic function \(f\), with non zero \(f'(0)\), satisfying the following geometric condition. Any Jordan curve on the surface \(R(f)\) with projection on the ``pencil circles'' at \(f(0)\), \(k\), \(t\), \(|t| > 0\), defined via \(\mathrm{Im\,}\exp(ik)/(w - f(0)) = 1/t\), that contains \(f(0)\) and does not pass through the ramification point of \(R(f)\) must be a univalent curve, i.e., one-sheeted.