How to describe $x \log (\log x)$?

Question

From what I understand, $x \log x$ is usually referred to as 'quasilinear' because it grows increasingly linear as $x$ increases. I was wondering then, if $x \log \log x$ could also be described as 'quasilinear', as it also grows increasingly linear as $x$ increases, and how to describe the difference between $x \log x$ and $x \log \log x$, seeing as both look somewhat similar when graphed (initial 'exponential'looking curve which grows straighter & straighter)

Answer 1

From Wikipedia I see that the definition of quasi-linear is generally functions in $O(n \log(n))$, or in $O(n (\log)^k)$. Since we have that $\log n $ is in $O(n)$, we also get that $n \log(\log(n))$ is in $O(n \log n)$. So technically "quasi-linear" would be fine to characterize it using this definition. Big $O$ notation is somewhat off a rough measure since it is only an upper bound though, so if you have a particular application or problem in mind you could look at the complexity a big more closely.

Answer 2

The domain of this function $f(x)=x \ln \ln x ~$ is $(1,\infty)$ Range is $(-\infty,\infty)$. This is an increasing function as $f'(x)=\ln \ln x +\frac{1}{\ln x}>0.$ This function has got a pint of inflection at $x=e$ as $f''(x)=\frac{1}{x}\left(\frac{1}{\ln x}- \frac{1}{\ln^2 x} \right).$ In $x \in (1,e)$ the curvature is negative and positive for $x \in (0,\infty)$. Se the below figure for $f(x)$

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