# Improving time complexity from O(log n/loglog n) to O((log ((nloglog n)/log n))/loglog ((nloglog n)/log n))

Suppose I have an algorithm whose running time is $$O(f(n))$$ where $$f(n) = O\left(\frac{\log n}{\log\log n}\right)$$

And suppose I can change this running time in $$O(1)$$ steps into $$O\left(f\left(\frac{n}{f(n)}\right)\right)$$, i.e. I can get an algorithm whose running time is $$O(g(n)) = O\left(\frac{\log\frac{n}{\frac{\log n}{\log\log n}}} {\log\log\frac{n}{\frac{\log n}{\log\log n}}}\right) = O\left(\frac{\log\frac{n\log\log n}{\log n}} {\log\log\frac{n\log\log n}{\log n}}\right)$$.

I’m pretty sure that $$g(n) < f(n)$$ for big enough $$n$$ (by using wolfram alpha) but wasn’t able to prove it.

My questions are:

1. Is $$g(n) < f(n)$$ in fact true (starting from some n)?

2. Is $$g(n)$$ asymptotically better the $$f(n)$$, i.e. is $$g(n) = o(f(n))$$

3. Assuming this is asymptotically better, I can do this step again and further improve the running time of the algorithm. Meaning that in 1 more step I can make my algorithm run in time of $$O\left(\frac{n}{f\left(\frac{n}{f(n)}\right)}\right)$$, and I can repeat this process as many times as I want. How many times should the process be repeated to get the best asymptotically running times and what will it be? obviously repeating it $$O(f(n))$$ times will already have a running time of $$O(f(n))$$ only for the repetition of this process and will not improve the overall algorithm complexity.