Big-O Asymptotic growth rate ordering functions

Question

This is what I have ordered the functions in increasing order of asymptotic growth rates. Also, I have simplified some functions by applying logarithmic rules.

1. log( log n )
2. sqrt( log n )
3. log n^3 (which is equal to log n)
4. n^2/3
5. 2^logn (which is equal to n)
6. n log n
7. n^2

Is this order correct? Or am I missing something?

Answer 1

O( A(n) ) < O( B(n) ) holds iff A(n) / B(n) approaches 0 when n goes to infinity.

You can check your table here: https://www.wolframalpha.com/input/?i=limit+log%28log%28n%29%29+%2F+sqrt%28log%28n%29%29%2C+n+to+infinity

For example

log(log(n)) / sqrt(log(n)) -> 0 for n -> inf

Hence O(log(log(n)) < O(sqrt(log(n)).