  We establish some estimates of the the angular derivatives from below for
holomorphic self-maps of the unit disk at one and two fixed points of the unit
circle provided there is no fixed point inside the unit disk. The results
complement Cowen-Pommerenke and Anderson-Vasil'ev type estimates in the case of
univalent functions. We use the method of extremal length and propose a new
semigroup approach to deriving inequalities for holomorphic self-maps of the
disk which are not necessarily univalent using known inequalities for univalent
functions. This approach allowed us to receive a new Ossermans type estimate as
well as inequalities for holomorphic self-maps which images do not separate the
origin and the boundary.
