I am trying to find an analytical expression of the minimum of
$$ f_n(x) = \frac{2x}{n^2+n}\log(x) - \frac{2x+2}{n^2+3n+2}\log(x+1) $$
when $x\in [1;n]$
I used to think from graphing it that this function is convex, but it is not. The second derivative is smaller than 0 at times. I tried to differentiate it anyway. Its derivative is : $$2\frac{(n +2 )\log{\left (x \right )} - 2n \log{\left (x + 1 \right )} + 2}{n \left(n^{2} + 3 n + 2\right)} $$
This is null when : $$(n+2)\log(x)+2 = 2n \log(x+1)$$
I read things about logarithmic equations, and I suspect that maybe the Lambert W function should appear somewhere, but I was unable to reduce this to the form $Y = Xe^X$.
I also saw that $f_n$ converges to the null function. I don't know if that's useful.
Any help (or even just pointers to relevant litterature) on the following topics would be appreciated:
- How to find the analytical expression of the minimum of $f_n$ ?
- How to solve the last equation ?
- Is there such a thing as 'almost convex' ? How can I convince people that I have found the true minima ?
Thanks in advance !