Showing that equation has unique real solution

Question

Let $epsilon_i$ be the internal energies of a microstate. We are supposed to show that the equation $$mathscr E = frac{sum_{i=1}^{N}epsilon_imathrm e^{-beta epsilon_i}}{sum_{i = 1}^{N}mathrm e^{-beta epsilon_i}}$$ has a unique real solution for $beta$. We can assume the following restriction:  $$text{min}(epsilon_i) leq mathscr Eleq frac{sum_{i = 1}^{N}epsilon_i}{N}.$$

Alright, I am not quite sure yet how to finally solve this task, but here are two ideas of mine:

I first multiplied by the denominator and applied the logarithm on both sides, but since there is no rule for $ln(a+b)$, this isn't fruitful, I guess. 
Maybe it suffices to show that $mathscr E(beta)$ is a strictly monotonic function? But then the problem is that the first derivative looks 'horrible'.

Here is the first partial derivative: $$frac{partialmathscr E}{partial beta} = frac{  sum_i mathrm e^{-beta epsilon_i} cdot left(-sum_i epsilon_i^2 mathrm e^{-beta epsilon_i} right) - sum_iepsilon_i mathrm e^{-betaepsilon_i} cdot left(-sum_iepsilon_i mathrm e^{-betaepsilon_i}right)  }{left(sum_i mathrm e^{-betaepsilon_i}right)^2} = frac{  sum_i mathrm e^{-beta epsilon_i} cdot left(-sum_i epsilon_i^2 mathrm e^{-beta epsilon_i} right) + left(sum_iepsilon_i mathrm e^{-betaepsilon_i}right)^2 }{left(sum_i mathrm e^{-betaepsilon_i}right)^2}$$

I am sure that ansatz (2) must work, but how I don't know yet. I guess that the trick is now to show that the first derivative is strictly positive or negative.

EDIT: I found the following solution online (under a password-protected site):

I don't even understand the second line, where does $U_i^2$ suddenly come from? ("Betrachte Zähler" is German for "Consider the numerator".)

Hans Moleman · Answer

I have a little bit of a glib argument. You want to prove that for $mathcal{E}$ in a certain range, there exists a $beta$ such that
$$
mathcal{E} = langle E rangle
$$
where the expectation value of course depends on $beta$.
You have computed that

$$- frac{d langle E rangle}{d beta} = langle E^2 rangle - langle E rangle^2.$$

But in any statistical average, such a difference is positive, since
$$ langle E^2 rangle - langle E rangle^2 = langle (E - langle E rangle)^2 rangle geq 0$$
so it follows that $langle E rangle$ is monotonically decreasing with $beta$. (It can only be constant if there is no variance in $E$ i.e. all $epsilon_i$ are identical.)

Finally you have to check the limits of $langle E rangle$ as $beta to 0,infty$ and you'll see that they coincide with the question.

A mathematician could have used a different argument: starting from the fact that $x mapsto x^2$ is a convex function, she would appeal to Jensen's inequality. This is more general and applies to any convex or concave function, not just the function $x mapsto x^2$.

Showing that equation has unique real solution

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