# Maximum value of a function with iterated logarithms

Consider this function $f(x)=-x\log\log(bx)+x\log\log\log(bx)+ax$, where $a$ and $b$ are positive constants, and the logarithms are 2-based.

Is it possible to find the maximum value (or even with approximation) of $f(x)$, in terms of $a$ and $b$, for $x>\frac{16}{b}$?

Note, it can be shown that $f(x)$ is concave for $x>\frac{16}{b}$.

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Why would you want to do that? – Will Jagy Feb 6 '12 at 8:37

If you call that maximum value $M(a,b)$, you can first of all get rid of $b$ since $M(a,b)= M (a,1) / b$. And $M(a,1) = g ^ \star (a)$, the Legendre transform of the convex function $g(x):= x\log \log x -x\log\log\log x$ for $x \ge 16$, and $g=\infty$ for $x < 16$. Then, you may at least estimate $g ^ \star$ integrating $(g^\star)'=(g')^{-1}$. Note that you may also consider the somehow more natural definition of $g$ with finiteness domain $]2,\infty[$, which seems still convex.
Thanks for you suggestion. However, I think to find the inverse mapping $(g^')^{-1}$ is even harder. – Allen Feb 6 '12 at 17:50