Boundary distortion estimates for holomorphic maps

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

On the stability problem in the O(N) nonlinear sigma model

The stability problem for the O(N) nonlinear sigma model in the 2+\epsilon dimensions is considered. We present the results of the 1/N^{2} order calculations of the critical exponents (in the 2<d<4 dimensions) of the composite operators relevant for this problem. The arguments in the favor of the scenario with the conventional fixed point are given.Comment: 9 pages, revtex, 1 Postscript figur