Treffer: a multiplier estimate of the schwarzian derivative of univalent functions: A multiplier estimate of the Schwarzian derivative of univalent functions

For an analytic function \(f\), univalent on the unit disk \(\mathbb D\) in the complex plane, the Schwarzian derivative \({\{f,z\}={\left(\frac{f''(z)}{f'(z)}\right)}'-\frac{1}{2}{\left(\frac{f''(z)}{f'(z)}\right)}^2 }\) is known to have analytic and geometric interpretations. For example, if \(f\) is a quasiconformal mapping of the plane with complex dilatation \(\mu\) and conformal on the unit disk, then the norm of the Schwarzian derivative of the restriction \(\tilde{f}\) of \(f\) to \(\mathbb D\) is related to the infinity norm of \(\mu\) by \( \|\{\tilde{f},z\}\|\leq 6\|\mu\|_{\infty}\). In particular, \(\| \{\tilde{f},z\}\|\) can be interpreted as a measure of how much \(f\) deviates from a Möbius transformation of the plane [see, for example, \textit{O.~Lehto}, ``Univalent functions and Teichmüller spaces'' (Graduate Texts in Mathematics 109, Springer--Verlag, New York et al.) (1987; Zbl 0606.30001). In the paper under review, the author obtains another type of norm estimate for the Schwarzian derivative as a mapping between certain weighed Bergman spaces. For each \(\alpha>1\), \({\mathcal H}_{\alpha}\) is the (weighed) Bergman space of all analytic functions \(g\) in \(L^2(\mathbb D,dA_{\alpha}(z))\), where \(dA_{\alpha}(z)=(\alpha-1)\left(1-| z| ^2\right)^{\alpha-2}dA(z)\) and \(dA(z)\) is the normalized area measure in \(\mathbb D\). The space \({\mathcal H}_\alpha\) is thus a Hilbert space with the above \(L^2\) norm. The main result states that if \(f\) is analytic and univalent in the unit disk, then for any \(\alpha>1\) and \(g\in{\mathcal H}_{\alpha}\), \(\| \{f,z\}g(z)\| _{\alpha+4}^2\leq 36\frac{\alpha+1}{\alpha-1}\| g\| _{\alpha}^2\). Interestingly, this estimate leads to improvements of bounds for the integral means spectrum of derivatives of univalent functions. In particular, upper bounds for the universal integral means spectrum \(B(t)\) at \(t=-1,-2\) are improved from \(B(-1)\leq 0.601\) to \(B(-1)\leq 0.4195\) and \(B(-2)\leq 1.547\) to \(B(-2)\leq 1.246\).