Expectation value of spin in an Ising lattice

[phagist_]

Homework Statement

I have to show that (the question says deduce from the fact that magnetization is monotonically increasing and a concave function for h>0)
$$\left< \sigma^2_{j} \right> - \left \sigma_j \right>^2 \geq 0$$
and $$\left< \left( \sigma^_{j} \right> - \sigma_j \right)^2 \right> \geq 0$$

Homework Equations

This in the context of a spin-1 Ising paramagnet in external field $$h$$
I have the fact that the magnetization is monotonically increasing and a concave function for $$h > 0$$

The Attempt at a Solution

I know the expression for an arbitrary observable;
Basically I have;
$$\left< \sigma_j \right> = \frac{\sum_{\sigma} \sigma_j e^{\beta h \sum_{j=1}^{N} \sigma_j}}{Z}$$

I'm just not sure how to use the concavity and the fact that m is monotonically increasing to find this result.

Any help is greatly appreciated.

 

[phagist_]

never mind, I figured it out.