[en] We propose a method in order to derive bounds on sets of threshold expansion parameters of the Compton scattering amplitudes in terms of the fixed (zero) momentum transfer cross section above the pion photoproduction threshold. The results represent strong and weak approximations to the optimal bounds. On the examples computed in detail it turns out, however, that the weak approximations are in fact the exact results (optimal bounds). If this is a general property is not decided in this paper. The mathematical background of the method, which should be applicable also to other problems, consists of interpolation theory for analytic functions and of boundary value problems for analytjc matpix-valued functions. (author)