Suppose we have a compact metrizable space $X$ and two metrics $d,h$ such that there is $a\in (0,1)$ and $c>0$ with $$h(x,y)\leq d(x,y)\leq c h(x,y)^a.$$
Can we deduce that $d,h$ are Holder equivalent for some $b\in (0,1)$?
Meaning $$c_1h(x,y)^b\leq d(x,y)\leq c_2 h(x,y)^b$$ for some $c_1,c_2>0?$
It seems a bit weird to hold whenever $d(x,y)<1$.
Thank you!