  In this work, we obtain the following new results.
  1. Given a sequence $D=((h_1,s_1), (h_2,s_2) ..., (h_n,s_n))$ of number
pairs, where $s_i>0$ for all $i$, and a number $L_h$, we propose an O(n)-time
algorithm for finding an index interval $[i,j]$ that maximizes
$\frac{\sum_{k=i}^{j} h_k}{\sum_{k=i}^{j} s_k}$ subject to $\sum_{k=i}^{j} h_k
\geq L_h$.
  2. Given a sequence $D=((h_1,s_1), (h_2,s_2) ..., (h_n,s_n))$ of number
pairs, where $s_i=1$ for all $i$, and an integer $L_s$ with $1\leq L_s\leq n$,
we propose an $O(n\frac{T(L_s^{1/2})}{L_s^{1/2}})$-time algorithm for finding
an index interval $[i,j]$ that maximizes $\frac{\sum_{k=i}^{j}
h_k}{\sqrt{\sum_{k=i}^{j} s_k}}$ subject to $\sum_{k=i}^{j} s_k \geq L_s$,
where $T(n')$ is the time required to solve the all-pairs shortest paths
problem on a graph of $n'$ nodes. By the latest result of Chan \cite{Chan},
$T(n')=O(n'^3 \frac{(\log\log n')^3}{(\log n')^2})$, so our algorithm runs in
subquadratic time $O(nL_s\frac{(\log\log L_s)^3}{(\log L_s)^2})$.
