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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 secondconjecture that the Narayana transformation preserves the log-convexity. Basedon a formula of Br\"and$\mathrm{\acute{e}}$n which expresses the $q$-Narayananumbers as the specializations of Schur functions, we derive several symmetricfunction identities using the Littlewood-Richardson rule for the product ofSchur functions, and obtain the strong $q$-log-convexity of the Narayanapolynomials and the strong $q$-log-concavity of the $q$-Narayana numbers.

Schur Positivity and the $q$-Log-convexity of the Narayana Polynomials