  In recent years, Sun has proposed numerous conjectures regarding the
log-concavity of root sequences $\{\sqrt[n]{a_n}}_{n\geqslant 1}$. We establish
criteria for the asymptotic log-concavity of $\{\sqrt[n]{a_n}}_{n\geqslant 1}$
and the asymptotic ratio log-convexity of $\{\sqrt[n]{a_n}}_{n\geqslant 1}$ for
$P$-recursive sequences $\{\sqrt[n]{a_n}}_{n\geqslant{0}}$. Additionally, by
the aid of symbolic computation, we present a systematic approach to determine
the explicit integer $N$ such that the sequence
$\{\sqrt[n]{a_n}}_{n\geqslant{N}}$ is log-concave and the sequence
$\{\sqrt[n]{a_n}}_{n\geqslant N}$ is ratio log-convex.
