  Two particular ratios related to mesons are proposed for the study of the
conformal window in $SU(3)$ gauge theory and fundamental fermions. Lattice and
other studies indicate that the lower end, $N_f^*$, is at around 7 - 13 flavors
which is a wide range without a clear consensus. Here we propose the decay
constant to mass ratios of mesons, $f_{PS,V} / m_V$, as a proxy since below the
conformal window lattice studies have shown that they are largely
$N_f$-independent while at the upper end of the conformal window they are
vanishing. The drop from the non-zero constant value to zero at $N_f = 16.5$
might be indicative of $N_f^*$. We compute $f_V / m_V$ to N$^3$LO and $f_{PS} /
m_V$ to NNLO order in (p)NRQCD. The results are unambiguously reliable just
below $N_f = 16.5$, hence the results are expanded \'a la Banks-Zaks in
$\varepsilon = 16.5 - N_f$. The convergence properties of the series and
matching with the non-perturbative infinite volume, continuum and chiral
extrapolated lattice results at $N_f = 10$ suggest that the perturbative
results might be reliable down to $N_f = 12$. A sudden drop is observed at $N_f
= 12$ and $N_f = 13$ in $f_V / m_V$ and $f_{PS} / m_V$, respectively.
