A question involving directional derivatives and differential inequalities 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*}
\frac{\partial C(a,a)}{\partial u} + \frac{\partial C(a,a)}{\partial v} \geq \frac{C(a,a)}{a}
\end{align*}
where $C$ is a copula and $0 < a \leq 1$. 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\\\\
\displaystyle \frac{\partial^{2}C}{\partial u\partial v} = \frac{\partial^{2} C}{\partial v\partial u}
\end{cases}
\end{align*}
This is all that I know for the moment. I tested the given property for $C(u,v) = \min\{u,v\}$, $C(u,v) = uv$ and $C(u,v) = \frac{uv}{1-(1-u)(1-v)}$ and it has worked well so far. Based on such considerations, could someone provide a partial or full answer to my question?
 A: A bivariate copula is the distribution function $C$ of a random variable $(X,Y)$ on $[0,1]\times[0,1]$ where the marginal distributions of $X$ and $Y$ are both uniform (Sklar's Theorem).
That is, there exists a random variable $(X,Y)$ for which (for all $0\le u,v\le 1$)
$$C(u,v) = \Pr(X\le u,\, Y\le v) \tag{1}$$
and
$$\Pr(X\le u) = u,\ \Pr(Y\le v)=v.$$
The question compares a derivative of $C$ to the values of $C.$  The former is local information while the latter is global (in the sense that $C(u,v)$ accumulates all the probability density of smaller values of $X$ and $Y$ but the first derivatives provide the rates at which $C$ is changing at the point $(u,v)$).  We should therefore expect no such inequality to hold generally.
A counterexample would be a case where $C(u,u)$ is large but its derivatives are small. In all copulas, $C(u,u)$ is small for small $u$ and many of them have small derivatives for small $u,$ too. One approach, then, is somehow to shift the smaller parts of a copula over to large values of $u,$ without changing its derivatives, because the accumulated probability at large $u$ will produce the counterexample.
This motivates the following construction of the "$a$-sum" $C\oplus_a D$ of two copulas $C$ and $D.$  For any $0\lt a\lt 1$, we will rescale $C$ to the rectangle $[0,a]^2$ and shift and rescale $D$ to the rectangle $[1-a,1]^2.$  Provided we extend the domain of definition of the copulas to include all real numbers (which formula $(1)$ automatically does), linear transformations easily accomplish this:
$$(C\oplus_a D)(u,v) = a C\left(\frac{u}{a}, \frac{v}{a}\right) + (1-a) D\left(\frac{u-a}{1-a}, \frac{v-a}{1-a}\right).$$
It is straightforward to check that this defines a copula and, if all first partial derivatives of $C$ and $D$ are less than $1$ (an additional condition given in the question), then all first partials of the $a$-sum are less than $1,$ too.
To illustrate, let $C(u,v)=D(u,v)=uv$ be the "independence copula" (representing independent uniform random variables).  Here is a filled contour plot of $C\oplus_{0.78}D:$

(The vertical and horizontal lines at coordinate $a=0.78$ show where one of the first partial derivatives is undefined.  The remaining black curves are the contours of $C.$)
To compare $C$ to the sum of its first partial derivatives, I have shaded the colors according to that sum, as shown in the legend, so that darker areas have smaller derivatives.
Note the region near $(a,a)=(0.78,0.78):$ above and to the right of that point, the first partial derivatives are reset to zero, as indicated by the dark shading.  However, the value of $C$ in that region cannot be any less than $a,$ because a proportion $a$ of all the probability lies (by construction) in the square $[0,a]^2$ below and to the left.
In the next plot I have shown both the values of $C(u,u)$ (in red) and the sum of the first partial derivatives (in black).

Right at $a=0.78$ the sum of first partials drops from $2$ back to $0,$ but (of course) the value of $C$ cannot decrease.  In the region from $0.78$ to approximately $0.87,$ the sum of first partials is less than $C(u,u),$ so a fortiori it is less than $C(u,u)/u.$ That's a counterexample.

Note that the partial derivatives in this counterexample are undefined at some points.  Since the question explicitly entertains such possibilities as $C(u,v)=\min(u,v),$ which have undefined first partials, I have understood that not to be a concern.

An interesting counterexample is afforded by $C=W\oplus_{1/2}W$ where $W$ is the Fréchet–Hoeffding minimum copula
$$W(u,v) = \max(0, u+v-1).$$
$W$ is the distribution function of the perfectly anticorrelated uniform random variable $(X,1-X).$  There is an entire triangular area where $C$ has attained a value of $1/2$ but is flat (both partial derivatives are zero):

This example is noteworthy because the corresponding random variable $(X,Y)$ is positively correlated: the Pearson correlation coefficient is $1/2.$
Reference
Roger B. Nelsen (2005), An Introduction to Copulas.  Second Edition, Springer.
A: positive correlation copulas
I guess that most copulas with sufficient positive correlation should work

Geometric intuition
These large correlation distributions look like a sort of mountain where the line along $a,a $ is a sort of rib. There is a steep rise to $a,a $ after which it is sort of flat.
You can also see it as: you do not increase much probability when moving either $u$ or $v$ because most mass is on the axis. When the correlation is highly positive then $$P (u \le a,v \le a) \approx P (u \le a) = P (u \le a, v \le 1)$$ So once you reach $a,a$ there is not much increase (small slope) from $P(u \le a, v\le a)$ to $P (u \le a, v\le 1)$
See this image from Wikimedia

The image on the right, fully correlated, has even zero derivatives in direction $u$ and $v$ (with a bit of imagination and ignoring that the derivative is not well defined there).

For instance, take the Ali-Mikhail-Haq copula
$$C (u,v) = \frac {uv}{1-\theta (1-u)(1-v)} $$
with derivatives
$$C (u,v)_u = \frac{\theta(v-1)v+v}{(1-\theta (1-u)(1-v))^2}$$
$$C (u,v)_v = \frac{\theta(u-1)u+u}{(1-\theta (1-u)(1-v))^2}$$
and say we take the point $a=u=v=0.5.$
Then
$$C (0.5,0.5)_u = \frac {C (0.5,0.5)}{0.5} \times \left (0.5 \frac {2+\theta}{1-0.25 \theta} \right)$$
and when $\theta <-0.8$ then you have that $C (0.5,0.5)_u +C (0.5,0.5)_v > \frac {C (0.5,0.5)}{0.5}.$
