Recursion and growth estimates in renormalizable quantum field theory

In this paper we show that there is a Lipatov bound for the radius of convergence for superficially divergent one-particle irreducible Green functions in a renormalizable quantum field theory if there is such a bound for the superficially convergent ones. In the nonnegative case the radius of convergence turns out to be min{rho, 1/b(1)}, where rho is the bound on the convergent ones, the instanton radius, and b(1) the first coefficient of the beta-function, while in general it is bounded by the above.