# Gradient estimate for convex function

Consider $\varphi:\mathbb{R}^n\rightarrow \mathbb{R}$ a convex, twice differentiable function with the gradient $\nabla \varphi$ Lipschitz-continuous. Suppose the function achieve a minimum in $\mathbb{R}^n$. We can write the following gradient system: $$\begin{cases} \dot{x}(t) = -\nabla\varphi(x(t))\\ x(0) = x_0 \in \mathbb{R}^n \end{cases}$$ From a much more general case it is known that the following estimate is true: $$\|\nabla \varphi(x(t))\|\leq \frac{C}{t}$$ for a certain constant $C$. I would like to prove this inequality without using the general case, but I haven't found anything yet. How can I prove it?

• Probably some assumption is missing. For example, in the case $\varphi(x) = x$ I do not see such gradient decay estimate. – Rigel Sep 15 '18 at 11:44
• Correct, the function must achieve a minimum in R^n – Paolo Sep 15 '18 at 11:46