I would like to know the sign of the following term in general. I tried approximation for $\log$ function and it had negative sign. Is there any $m\_0$ such that for all $n>m>m\_0$, the following function is positive or as I get it is always negative in its domain.


$$ f(m,n)=\frac{\log\log\log m}{\log\log\log n} \centerdot \frac{\log\log n}{\log\log m}-\left(1+\frac{\log(\frac n m)}{(\log n)\log\log m}\right)$$


where $n$ is greater than $m$.


thanks



determining sign of function containing logarithm.
This is my second response, after the clarifying remark from the asker (see below).


Let us assume that $m>e^e$ so that $\log\log m>0$. Then there is $m\_0>m$ such that $f(m,n)<0$ for any $n\in(m,m\_0)$. To see this claim, let $\epsilon>0$ be small, and consider $n$ such that 
$$ \log n=(\log m)^{1+\epsilon}. $$ Then
$$ \frac{\log\log\log m}{\log\log\log n} \centerdot \frac{\log\log n}{\log\log m} = \frac{(1+\epsilon)\log\log\log m}{\log\log\log m+\log(1+\epsilon)} $$
$$=1+\epsilon-\frac{\epsilon}{\log\log\log m}+O(\epsilon^2)$$
while
$$ \frac{\log(\frac n m)}{(\log n)\log\log m}= \frac{1-(\log m)^{-\epsilon}}{\log\log m} = \frac{\epsilon\log\log m+O(\epsilon^2)}{\log\log m}=\epsilon+O(\epsilon^2).$$
Therefore
$$ f(m,n) = -\frac{\epsilon}{\log\log\log m}+O(\epsilon^2)$$
which is negative if $\epsilon>0$ is sufficiently small in terms of $m$.


**P.S.** My original response contained the more obvious claim that $f(m,n)>0$ when $n$ is sufficienly large in terms of $m$. Hence, for any $m>e^e$, there exist negative and positive values of $f(m,n)$ with $n>m$.


