How to calculate Big O of $T(n) = aT(n^b) + f(n)$?

Question

I'm a student studying Big O. I know that we can solve $T(n) = aT(\frac{n}{b}) + f(n)$ by compering $n^{\log_b{a}}$ to $f(n)$ or

$O(n^{\log_b{a}} + f(n))$

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Today I was faced with $T(n) = T(\sqrt n) + 1$ and I think it is $O(\log\log n)$. and I was faced with this too, $T(n) = T(\sqrt n) + O(\log\log n)$ that I think it is $O(\log^2\log n)$.

I'm wondering what is the formula (or method) to solve any kind of this type of problems (my guess is $O(f(n)a^n\log\log n)$). so:

How to calculate Big O of $T(n) = aT(n^b) + f(n)$ with $0<b<1$?

Thanks in advance.