Simplification of Product

Question

I am loking for alternative ways to write the ratio $$ f(a_1, \dots, a_N) = \frac{\prod_i^N a_i}{\min\{ a_1 \dots a_N \}} $$ where $0 \le a_i \le 1$.

For $N=1$, it reduces to $\max(a_1, a_2)$. However, I am looking for alternative expressions.

Answer 1

For $N=1$ you get $f(a_1)=\dfrac{\prod_{i=1}^1a_i}{\max_{i=1}\{a_i\}}=\dfrac{a_1}{a_1}=1$.

For $N=2$ you would get $f(a_1,a_2)=\dfrac{a_1a_2}{\max\{a_1,a_2\}}$.

What do you not like about the proposed way of writing this ratio? I think it is the most easy to understand/work with. I do not think that there is much you can do. You could evaluate the maximum by this formula:

$\max\{x_1,x_2\}=\dfrac{x_1+x_2+|x_1-x_2|}{2}$ and $\max\{x_1, x_2,\dotso, x_n\}=\max\{x_1, \max\{x_2,\dotso, x_n\}\}$

And then iterate the formula. But that is just ugly.