  We work out and discuss the Minkowski version of Fractional Analytic
Perturbation Theory (FAPT) for QCD observables, recently developed and
presented by us for the Euclidean region. The original analytic approach to
QCD, initiated by Shirkov and Solovtsov, is summarized and relations to other
proposals to achieve an analytic strong coupling are pointed out. The developed
framework is applied to the Higgs boson decay into a b\bar{b} pair, using
recent results for the massless correlator of two quark scalar currents in the
\bar{MS} scheme. We present calculations for the decay width within MFAPT
including those non-power-series contributions that correspond to the
O(\alpha_{s}^{3})-terms, taking also into account evolution effects of the
running coupling and the b-quark-mass renormalization. Comparisons with
previous results within standard QCD perturbation theory are performed and the
differences are pointed out. The interplay between effects originating from the
analyticity requirement and the analytic continuation from the spacelike to the
timelike region and those due to the evolution of the heavy-quark mass is
addressed, highlighting the differences to the conventional QCD perturbation
theory.
