  We prove three main conjectures of Berkovich and Uncu (Ann. Comb. 23 (2019)
263--284) on the inequalities between the numbers of partitions of $n$ with
bounded gap between largest and smallest parts for sufficiently large $n$.
Actually our theorems are stronger than their original conjectures. The
analytic version of our results shows that the coefficients of some partition
$q$-series are eventually positive.
