Very based question regarding expression transformation.

$$\log p_i \leq \log i + 2 \log\log i $$ $$ p_i \leq i(\log i)^2$$

I am not getting this transformation. So from what I try:

$$\log p_i \leq \log (2i \log\log i) $$ $$ p_i \leq 2i \log\log i$$ $$ p_i \leq 2i \log^2 i$$ $$ p_i \leq \log^2 i^{2i}$$

And I am stuck...


We have $$log(p_i)\leq log(i)+2\log(log(i))$$ $$log(p_i)\leq log(i)+\log(log(i)^2)$$ $$log(p_i)\leq \log(i\log(i)^2)$$ $$p_i\leq i\log(i)^2$$

Here the second inequality follows since $a\log(b)=\log(b^a)$, and the third since $\log(a)+\log(b)=\log(ab)$. The last inequality holds since th logarithm is an increasing function.


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