Characterisation of the subdifferential of a $\lambda$-convex function.

Question

If $F$ is convex, then the inequality that characterizes the subgradient is $$F(y)\ge F(x)+\langle p,y-x\rangle$$ and if F is $\lambda$-convex (i.e. $F(x)-\frac{\lambda}{2}\|x\|^2$ is convex), the inequality that characterises the subgradients is $$F(y)\ge F(x)+\frac{\lambda}{2}|y-x|^2+\langle p,y-x\rangle.$$ I don't understand how we got the characterization for the $\lambda$-convex function.