Does the opposite inequality of A.M ≥ G.M holds?

Question

Let $a,b>0$. Does there exists $C_1,C_2>0$ such that $a+b \leq C_1(ab)^{C_2}$ holds?

We know from A.M ≥ G.M that the opposite inequality holds for $C_1=2,C_2=½$

Answer 1

Assume such constants exist and then let $a\rightarrow 0$ then $$b=\lim_{a\rightarrow 0}a+b\leq\lim_{a\rightarrow 0}C_1(ab)^{C_2}=0$$ Contradiction!