Loewner equation, fixed points and angular derivative

The process of expansion (and contraction) of a simply connected domain is described by a family of conformal mappings onto a canonical domain (disc, half-plane, strip). The question of differentiability of the resulting family of conformal mappings depends on the canonical domain, normalization at fixed points, and the choice of the family parameter. In the case of boundary normalizations, the choice of parameter is closely related to the angular derivative at a fixed point. It is intended to discuss the results related to the influence of the angular derivative on the character of the holomorphic mapping, the choice of parameter and the differentiability of families of conformal mappings.