Iterated logarithm Big-O complexity

Question

I have 2 doubts:-

1) Is (log* n)^n = O((logn)!) ?

2) Which is bigger, log(log* n) or log*(logn) ?

Answer 1

For 2), you have log*(log n) = log*(n)-1 . Then let m = log*(n) . You have m-1 > log(m) for sufficiently large m .

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Hint :

For 1), let m = log*(n) . Then the LHS is m^n , and the RHS is the factorial of the logarithm of an exponential tower of height m , ie the factorial of an exponential tower of height m-1 .

Even disregarding the factorial, an exponential tower should grow much faster than a power.