Quadratic differentials and conformal invariants

We define a notion of conformal invariance associated with nested domains, suitable for characterizing higher-order information about mapping functions. We give an exposition of our results which yield an infinite-dimensional family of conformal invariants for nested hyperbolic simply-connected domains. Each invariant is specified by a quadratic differential which is admissible for the outer domain, and is strictly negative unless the inner domain is the outer domain minus trajectories of the quadratic differential. These invariants are furthermore monotonic. Using the aforementioned invariants, we show that one can obtain various classical estimates for bounded univalent functions, and in many cases extend them, by choosing particular quadratic differentials. We also explain the principles behind these results and their context within the literature. © 2017, Forum D'Analystes, Chennai.