  Acquaah and Konyagin showed that if $N$ is an odd perfect number where $N=
p_1^{a_1}p_2^{a_2} \cdots p_k^{a_k}$ where $p_1 < p_2 \cdots < p_k$ then one
must have $p_k < 3^{1/3}N^{1/3}$. Using methods similar to theirs, we show that
$p_{k-1}< (2N)^{1/5}$ and that $p_{k-1}p_k < 6^{1/4}N^{1/2}.$ We also show that
if $p_k$ and $p_{k-1}$ are close to each other than these bounds can be further
strengthened.
