This is a follow-up question to A question about copulas and directional derivatives. Since no answer was given, I am going to precise the definition of copula. I am interested in proving (or disproving) that \begin{align*} \langle\nabla C(a,a),(a,a)\rangle \geq C(a,a) \end{align*}

where $0\leq a \leq 1$ and $C$ is a bivariate copula. A bivariate copula is a function $C:[0,1]^{2}\rightarrow[0,1]$ such that \begin{align*} \begin{cases} C(1,t) = C(t,1) = t\\\\ C(0,t) = C(t,0) = 0\\\\ \displaystyle\frac{\partial^{2}C}{\partial u\partial v} = \frac{\partial^{2} C}{\partial v\partial u} \geq0 \end{cases} \end{align*}

It also satisfies the properties \begin{align*} \begin{cases} \max\{u+v-1,0\} \leq C(u,v) \leq \min\{u,v\}\\\\ \displaystyle 0 \leq \frac{\partial C}{\partial u} \leq 1\\\\ \displaystyle 0 \leq \frac{\partial C}{\partial v} \leq 1 \end{cases} \end{align*}

This is all that I know for the moment. I tested the given property for $C(u,v) = \min\{u,v\}$, $C(u,v) = uv$, $C(u,v) = \frac{uv}{1-(1-u)(1-v)}$ and $C(u,v) = u + v - 1 + \theta(1-u)(1-v)$ and it has worked quite well so far. Based on such considerations, could someone provide a partial or full answer to my question? It is worthy emphasizing that it is not an exercise. It makes part of my research on the theory of copulas.

**EDIT**

Sorry gentlemen, but I forgot to mention that $C$ must be symmetric.