Real functions with the property: $\ f(x_1)f(x_2) = f\left( \frac{x_1+x_2}{2} \right)^2 $ for all $\ x_1,\ x_2\in\mathbb{R}.\ $ Suppose $\ f:\mathbb{R}\to\mathbb{R}\ $ has the property:$\ f(x_1)f(x_2) = f\left( \frac{x_1+x_2}{2} \right)^2\ $ for all $\ x_1,\ x_2\in\mathbb{R}$.
I made some educated guesses and stumbled upon the fact that if $\ A,\alpha\in\mathbb{R},\ $ then $\ f(x) = A e^{\alpha x}\ $ satisfies this property.
I also realise that $\ f\ $ must be convex if $\ f>0\ $ and concave if $\ f<0$.
So now I'm wondering if any other functions satisfy the property, and if not, how to prove uniqueness of $\ f(x) = A e^{\alpha x}\ $ in satisfying the property. Edit: I want something stronger: to classify all the solutions to this functional equation.

As stochasticboy commented while I was drafting this:
Let $\lambda(x)$ be any discontinuous solution of the additive form of Cauchy's functional equation.  One checks trivially that $\lambda(2u)=2\lambda(u)$ for all $u$, equivalently $\lambda(u/2)=\lambda(u)/2$.  Now let $f(x)=\exp(\lambda(x))$,
so
$$ f(x)f(y)=e^{\lambda(x)+\lambda(y)}=e^{\lambda(x+y)} =e^{2\lambda((x+y)/2)}=\left(f\left(\frac{x+y}2\right)\right)^2,$$
taking $u=x+y$.
