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.