Suppose $a_1, ..., a_n \in \mathbb{N}$ are arbitrary integers. Is it possible to bound $$ A =\sum_{i=1}^n \sum_{j = i}^{i+a_i} \frac{1}{\sqrt{j}} $$ with either of the following: $$ B = c\sqrt{\sum_{i=1}^n a_i} $$ $$ C = c'\sqrt{\sum_{i=1}^n \frac{a_i}{i}} $$
for some arbitrary constants $c$, $c'$.
PS. I know that $\sum_{i=m}^n \frac{1}{\sqrt{i}} \leq 2 (\sqrt{n} - \sqrt{m})$, but not sure it is useful. The only thing I can get is: $$ A \leq \sum_{i=1}^{n} 2\sqrt{a_i + i} - 2\sqrt{i} = O\left(\sum_{i=1}^{n} \sqrt{a_i } \right) $$
Also using this we know that $$ \sum_{j = i}^{i+a_i} \frac{1}{\sqrt{j}} = O \left( \sqrt{ \frac{d_i}{i} } \right) \Rightarrow A = O\left(\sum_{i=1}^{n} \sqrt{ \frac{ a_i }{i}} \right) $$