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

TL;DR

This paper establishes sharp multiplier norm estimates for higher-order Schwarzian derivatives of the Koebe function, extending previous results and highlighting its extremal properties in univalent function theory.

Contribution

It provides explicit sharp estimates for these derivatives and demonstrates the Koebe function's extremal role among univalent functions.

Findings

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

The Koebe function remains extremal for certain higher-order Schwarzians.

Extension of previous results by Shimorin with explicit formulas and new theorems.

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.