$\min$ function properties

Question

I need your help to get an insight on the $\min$ function:

Let $a,b \in [0,\infty)$, then it is known that: $$ \frac{a+b}{2} \leq \max\left\lbrace a,b\right\rbrace \leq a+b$$

Is there a way to get the same inequality up to a multiplicative factor using the $\min$ function instead of $\max$ function?

If not, what inequalities can be derived using the $\min$ function?!

Please advise.

P.s. We are only interested in a lower bound!

Answer 1

if $$\max\{a,b\}=a$$ if $$a\geq b$$ we have $$\frac{a+b}{2}\le a$$ if $$b\le a$$ can you compute the other case?