Let $F:\mathbb R^n\to [0,\infty)$ be $C^1$ convex function such that $F(\alpha x)=\alpha^2F(x)$ for $\alpha>0$ and for $u,v$ are two positive differential functions with $u>0$. Then prove that $$F(\nabla v)\geq\frac{v^2}{u^2} F(\nabla u)+\langle\nabla F(\nabla u),\nabla v\rangle\frac vu-\langle\nabla F(\nabla u),\nabla u\rangle \frac{v^2}{u^2}$$ where $\langle,\rangle$ denotes the inner product.
I am trying to prove this one by using convexity but I can't get it.