  We study the $l^p$ norms of a class of weighted mean matrices whose diagonal
terms are given by $n^{\alpha}/\sum^{n}_{i=1}i^{\alpha}$ with $\alpha > -1$.
The $l^p$ norms of such matrices are known for $p \geq 2, (\alpha+1)p >1$ and
$1<p \leq 4/3, 1/p \leq \alpha \leq 1$. In this paper, we determine the $l^p$
norms of such matrices for $p \geq 1.35, 0\leq \alpha \leq 1$.
