$f_{c}^*$ means how many times taking function so it will become $'c'$
for f(n)=$\sqrt{n}$ ,c=2 $\rightarrow$ how many times taking root so it become 2
$n^{\frac{1}{2}}$ $\rightarrow$ $n^{\frac{1}{2^2}}$ $\rightarrow$ $n^{\frac{1}{2^3}}$....... $n^{\frac{1}{2^k}}$
$n^{\frac{1}{2^k}}$=2
k=$log \ logn$. $f_{c}^*$=$log \ logn$
2.f(n)=$\sqrt{n}$ ,c=2 it can not be converge .($n^{\frac{1}{2^k}}$=1)