  Perturbative QCD in mass independent schemes leads in general to running
coupling $a(Q^2)$ which is nonanalytic (nonholomorphic) in the regime of low
spacelike momenta $|Q^2| \lesssim 1 \ {\rm GeV}^2$. Such (Landau) singularities
are inconvenient in the following sense: evaluations of spacelike physical
quantities ${\cal D}(Q^2)$ with such a running coupling $a(\kappa Q^2)$
($\kappa \sim 1$) give us expressions with the same kind of singularities,
while the general principles of local quantum field theory require that the
mentioned physical quantities have no such singularities. In a previous work,
certain classes of perturbative mass independent beta functions were found such
that the resulting coupling was holomorphic. However, the resulting
perturbation series showed explosive increase of coefficients already at ${\rm
N}^4{\rm LO}$ order, as a consequence of the requirement that the theory
reproduce the correct value of the $\tau$ lepton semihadronic strangeless decay
ratio $r_{\tau}$. In this work we successfully extend the construction to
specific classes of perturbative beta functions such that the perturbation
series do not show explosive increase of coefficients, the perturbative
coupling is holomorphic, and the correct value of $r_{\tau}$ is reproduced. In
addition, we extract, with Borel sum rule analysis of the V+A channel of the
semihadronic strangeless decays of $\tau$ lepton, reasonable values of the
corresponding D=4 and D=6 condensates.
