  We prove various inequalities between the number of partitions with the bound
on the largest part and some restrictions on occurrences of parts. We explore
many interesting consequences of these partition inequalities. In particular,
we show that for $L\geq 1$, the number of partitions with $l-s \leq L$ and
$s=1$ is greater than the number of partitions with $l-s\leq L$ and $s>1$. Here
$l$ and $s$ are the largest part and the smallest part of the partition,
respectively.
