Rigorous control of the non-perturbative corrections to the double expansion in g and g(2)ln(g) for the phi(4)(3)-trajectory in the hierarchical approximation

We study the renormalization invariant trajectory of the phi(4)-perturbatio n of the free field fixed point in the hierarchical approximation. We param etrized it by a running phi(4)-coupling g with linear step P-function. We r igorously control the non-perturbative corrections to finite order approxim ants from double perturbation theory in g and g(2)ln(g). The construction u ses a contraction mapping for the extended renormalization group composed o f a hierarchical block spin transformation with a flow of g.