Are there stars with an average density greater than the central density? Are there likely to be stars for which the average density is greater than the density at the centre? Intuitively, I would say not, as density tends to decrease outwards, but ideally I would appreciate a clear mathematical justification for this.
$$\frac{dP}{dr} = - \rho g,$$
is the equation of hydrostatic equilibrium, where $\rho$ and $g$ are the local density and gravity, $P$ is pressure and $r$ is the radial coordinate. This can be rewritten as
$$\frac{d\rho}{dr} \frac{dP}{d\rho} = -\rho g.$$
Since $\rho$ and $g$ are positive numbers, the pressure gradient is negative.
For all types of matter $P = f(\rho)$, such that $dP/d\rho$ is also positive (i.e. pressure increases with density).
Thus $d\rho/dr$ is negative and the density increases towards the middle.
EDIT: This argument only obviously works if you ignore the dependence of pressure on other variables - such as the temperature or number of particles per unit mass in your gas. It takes more work to establish that $dP/d\rho&gt;0$ in a more general case. For example you might consider a situation where a sufficiently negative temperature gradient might counterbalance a positive density gradient.
i.e. if $P = f(\rho, T)$ then
$$ \frac{dP}{d\rho} = \frac{\partial P}{\partial \rho} + \frac{\partial P}{\partial T} \frac{dT}{d\rho}$$
and $dP/d\rho&lt;0$ if
$$ \frac{dT}{d\rho} &lt; -\frac{\partial P/\partial \rho}{\partial P/\partial T}$$
Leaving this as a placeholder for the moment.