If I know that for some function $f$, the following is true for $x, y \geq 0$:
$f(\log (x^a y^b)) \leq f(\log x)^a f(\log y)^b$
Can I make the claim that
$f(x^a y^b) \leq f(x)^a f(y)^b$
If I cannot generally, are there certain conditions under which I can make this claim?
No I don't think so. Taking log x = p & log y = q.
You can write it as
$f(ap + bq) \leq f(p)^a f(q)^b$
Now replace p and q by x and y.