It there a better upper bound for $\max_{1\leq i \leq n}(a_{i}*b_{i})$?

I would like to give an upper bound for $\max_{1\leq i \leq n}(a_{i}*b_{i})$ which is related to $\max_{1\leq i \leq n}(a_{i})$. I know that $\max_{1\leq i \leq n}(a_{i})\max_{1\leq i \leq n}(b_{i})$ can be one of them, but is there any better bound?