PQCD running couplings finite and monotonic in the infrared: When do they reflect the holomorphic properties of spacelike observables?

We investigate a large class of perturbative QCD (pQCD) renormalization schemes whose beta functions beta(a) are meromorphic functions of the running coupling and give finite positive value of the coupling a(Q(2)) in the infrared regime (freezing), a(Q(2)