  Using Schur positivity and the principal specialization of Schur functions,
we provide a proof of a recent conjecture of Liu and Wang on the
$q$-log-convexity of the Narayana polynomials, and a proof of the second
conjecture that the Narayana transformation preserves the log-convexity. Based
on a formula of Br\"and$\mathrm{\acute{e}}$n which expresses the $q$-Narayana
numbers as the specializations of Schur functions, we derive several symmetric
function identities using the Littlewood-Richardson rule for the product of
Schur functions, and obtain the strong $q$-log-convexity of the Narayana
polynomials and the strong $q$-log-concavity of the $q$-Narayana numbers.
