Show that $\frac1{\log\log(\gamma^r) (\log(\gamma^r))^\eta} \ge 1/r$.

Question

I want to show that $$\frac1{\log\log(\gamma^r) (\log(\gamma^r))^\eta} \ge 1/r$$ for $0 < \eta <1 $, and $r, \gamma$ sufficiently large integers. This inequality comes up in the middle of the proof of the law of the iterated logarithm, and I don't know how to show this. I was trying to show this using the basic log rules, but I have failed several times.

Could you give me some hint?

Answer 1

We need to show that $r^{1-\eta}\geq \ \ (log(r) + loglog(\gamma))(log(\gamma))^\eta$

But we know that $lim_{x\rightarrow\infty}x^{-a}log(x)=0 \ \ \forall \ \ a \gt 0$ Thus keeping $\gamma$ constant if we go on increasing $r$ the required inequality is obtained