It's a standard result that the number ω(N) of prime factors of N > 2 can be bounded above by$$ \omega(N) \;=\; \frac{\log(N)}{\log\log(N)} \big(1 + O\big(1/\log\log(N)\big)\big) \;.$$ Are tighter bounds known when N is logarithmically rough  that is, where for some fixed constant c > 0, N has no prime factors smaller than c log(N)?

The intuition is that $\log_{c\log N}N=\frac{\log N}{\log(c\log N)}=\frac{\log N}{\log c+\log\log N}$ which is asymptotically $\frac{\log N}{\log\log N}$. For this to work, we need that the kth prime for $k=\frac{\log N}{\log(c\log N)}+\pi(c\log N)\approx\frac{\log N}{\log\log N}$ to be 'close' to $c\log N$  actually, any constant multiple of $\log N$ will do. $p_k\approx\frac{\log N}{\log\log N}\log\frac{\log N}{\log\log N}\approx\log N$, so $\log_{p_k}N\approx\frac{\log N}{\log\log N}$, as desired. This could probably made explicit with Rosser's theorem and/or Dusart's various bounds on $p_n$ and $\pi(n)$. 

