Sharp multiplier estimates for the higher-order Schwarzian derivatives of the Koebe function

Abstract

In this note we study the multiplier norm estimates for the multiplication operators between weighted Bergman spaces, whose symbols are the higher-order Schwarzian derivatives of univalent functions. We establish sharp multiplier estimates for the higher-order Schwarzian derivatives of the Koebe function. This extends a related result by Shimorin. The proof of our new theorem relies on an explicit formula for the higher-order Schwarzian derivatives of the Koebe function and a recent theorem from our earlier work. We finally point out that the Koebe function is still the extremal function for certain higher-order Schwarzians of the univalent functions.