A nonlinear extremal problem for Bloch functions with applications to geometric function theory (Q1423847)

Let \(D\) be a domain in the complex plane equipped with a hyperbolic density \(\lambda_D\). For fixed \(C\in \mathbb R\) the author finds an upper bound for \(\sup_{w\in D}| g''(w)-Cg'(w)^2| /\lambda_D(w)^2\), where \(g\) is an analytic Bloch function in \(D\). This bound is best possible if \(D\) is simply connected. As a corollary the author obtains a sharp upper estimate for the norm of the Schwarzian derivative of an analytic function in a simply connected domain in terms of the norm of its pre-Schwarzian. This yields a nonlinear extension of a theorem of \textit{K-J. Wirths} [Arch. Math. 30, 606--612 (1978; Zbl 0373.30016)] and also a converse of a result of \textit{M. Chuaqui} and \textit{B. Osgood} [Comment. Math. Helv. 69, No. 4, 659--668 (1994; Zbl 0826.30013)]. As another corollary, the sharp upper bound of the third Taylor coefficient in the set of analytic functions \(f: \mathbb D \to \mathbb C\) satisfying \((1-| z| ^2)| f''(z)/f'(z)| \leq 1\) and normalized by \(f(0)=f'(0)-1=0\) is obtained.