Analytic properties of Poincaré series of spaces

Let X be a finite simply connected CW-complex of dimension nX such that π∗(ΩX)⊗Q is infinite dimensional. We prove that under some hypothesis on the Poincaré series of ΩX there exist positive numbers A and B such that for any positive integer k we haveAk ω-k⩽∑j=k+1k+nX-1 rkπj(ΩX)⩽Bk ω-kwhere ω is the radius of convergence of the Poincaré series of ΩX and ω-1>1.