  Let $n\geq2$ be an integer. In this paper, we study the convexity of the
so-called MacMahon's $q$-Catalan polynomials $C_n(q)=\frac1{[n+1]_q}\left[ 2n
\atop n \right]_q$ as functions of $q$. Along the way, several intermediate
results on inequalities are presented including a commentary on the convexity
of the generation function for the integer partitions.
