# Higher order Schwarzian Derivatives as multipliers between Bergman spaces

Abstract:

If $f :\mathbb D \to C$ is locally univalent, the Schwarzian derivative is defined as

$$\mathcal S(f) = \left( \frac{f’'}{f’} \right)’ - \frac 1 2 \left( \frac{f’'}{f’} \right)^2$$

If $f$ is univalent, this operator is conformally invariant and has a growth given by

$$(1-|z|^2)^2|\mathcal S(f)| \le 6, z \in \mathbb D$$

In this talk, we shall find higher order derivatives of this type and we shall obtain estimates on the norm of the operator between Bergman spaces consisting in the pointwise multiplication by these derivatives. As an application, we shall obtain information on the growth of the integral means of derivatives of conformal maps.

This talk is hosted by Universitat Autonoma de Barcelona. Link to the talk site.