I need your expertise in providing an upper bound for the following problem:
Given two sets of $n$ non negative real numbers $\{ a_i\}_{i=1}^n, \{ b_j\}_{j=1}^n$ where for every $i,j \in [n]$, $a_i,b_j \geq \frac{1}{2n}$, where $n > 1$, and $\sum\limits_{j=1}^n b_j \leq \sum\limits_{i=1}^n a_i$.
Can we prove that there exists $c \in \mathbb{R}_+$ such that $$ \sum\limits_{i=1}^n \frac{a_i}{b_i} \leq \frac{c\sum\limits_{i=1}^n a_i}{\sum\limits_{j=1}^n b_j} ? $$
If so, how we bound $c$ in any way?
Please advise and thanks in advance.
P.s. If the following can not be achieved, what are the necessary conditions that the sets of real number must apply in order to attain the desired bound?
Let $b_i=\frac{1}{2n}$ and $a_i=1$.
Hence, we need $$2n^2\leq c\cdot\frac{n}{\frac{1}{2}}$$ or $$c\geq n,$$ which says that needed $c$ does not exist.