The MacMahon $q$-Catalan is convex

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