Proof: From Apollonius's theorem: $4 m_{a}^{2}=2 b^{2}+2 c^{2}$
$$\begin{array}{l} -a^{2}, \text { we get } 4\left(m_{a}^{2}+m_{b}^{2}+m_{c}^{2}\right)=\left(2 b^{2}+2 c^{2}-a^{2}\right) \\ +\left(2 c^{2}+2 a^{2}-b^{2}\right)+\left(2 a^{2}+2 b^{2}-c^{2}\right)=3 a^{2}+ \\ 3 b^{2}+3 c^{2} . \end{array}$$

Obviously, the function $f(x)=3 x^{2}$ is a convex function in $(0,+\infty)$, so by Jensen's inequality we get:
$$3 a^{2}+3 b^{2}+3 c^{2} \geqslant 3\left[3 \cdot\left(\frac{a+b+c}{3}\right)^{2}\right]=$$
$4 s^{2}$, that is: $m_{a}^{2}+m_{b}^{2}+m_{c}^{2} \geqslant s^{2}$.