As an extension of Ozeki's inequality we give an inequality 
which estimates the difference
\begin{equation*}
  \sum_{k=1}^n  
p_k a_k^2\sum_{k=1}^n p_k b_k^2 - (\sum_{k=1}^n p_k a_k b_k)^2
\end{equation*}
derived from the weighted Cauchy-Schwartz inequality 
for $n$-tuples $a = (a_1, ..., a_n),\ b = (b_1, ..., b_n)$ 
and $p = (p_1, ..., p_n)$ of positive numbers under certain 
conditions. We discuss the upper bound of the difference 
not only in the general case but also in the special cases 
that $a$ and $b$ are monotonic in the opposite sense and 
in the same sense.