  We work out the interplay between lowest-order perturbative computations in
the 't Hooft coupling, $g^2=g^2_{YM} N$, operator mixing, renormalization-group
(RG) improved ultraviolet (UV) asymptotics of leading-order (LO)
nonplanar/planar contributions to $2$-point correlators, and nonperturbative
large-$N$ expansion of perturbatively massless QCD-like theories. As concrete
examples, we compute to the lowest perturbative order in $SU(N)$ YM theory the
ratios, $r_i$, of LO-nonplanar to planar contributions to the $2$-point
correlators in the orthogonal basis in the coordinate representation of the
gauge-invariant dimension-$8$ scalar operators and all the twist-$2$ operators.
We demonstrate that -- if $\frac{\gamma_0}{\beta_0}$ has no LO-nonplanar
contribution, with $\gamma_0$ and $\beta_0$ the one-loop coefficients of the
anomalous-dimension matrix and beta function respectively -- $r_i$ actually
coincides with the corresponding ratio in the large-$N$ expansion of the
RG-improved UV asymptotics of the $2$-point correlators, provided that a
certain canonical nonresonant diagonal renormalization scheme exists for the
corresponding operators. Contrary to the aforementioned scalar operators, for
the first $10^3$ twist-$2$ operators we actually verify the above conditions,
and we get the universal value $r_i=-\frac{1}{N^2}$. Hence, nonperturbatively
such $r_i$ must coincide with the UV asymptotics of the ratio of the glueball
self-energy loop to the glueball tree contribution to the $2$-point correlators
above. As a consequence, the universality of $r_i$ reflects the universality of
the effective coupling in the nonperturbative large-$N$ YM theory for the
twist-$2$ operators in the coordinate representation.
