Extending functional inequality from rectangles to parallelograms

Question

Let $f$ be a function defined on the unit square $R = [0,1]^2 \subseteq \mathbf{R}^2$ satisfying

• $f \geq 0$, $f(0,0) = 0$,
• $\frac{\partial{f}}{\partial{x}} \geq 0$, $\frac{\partial{f}}{\partial{y}} \leq 0$,
• $\frac{\partial^2{f} }{\partial{x}\partial{y}} \leq 0$.

The last condition is equivalent to the inequality $f(x_1,y_1) + f(x_2,y_2) \geq f(\min\left(x_1,x_2\right), \min\left(y_1,y_2\right)) + f(\max\left(x_1,x_2\right), \max\left(y_1,y_2\right))$ on any rectangle, which can be obtained by integrating $\frac{\partial^2{f} }{\partial{x}\partial{y}}$ over the rectangle.

If we label the vertices of the rectangle counterclockwise $v_1, \dots, v_4$, starting at the upper right, this is saying that $f(v_2) + f(v_4) \geq f(v_1) + f(v_3)$.

Does this property also hold for parallelograms inscribed in $R$ with wlog $v_1 = (1,1), v_3 = (0,0)$?

For $v_2 = (x_2,y_2)$, $v_4 = (x_1,y_1)$, the linear function mapping $R$ to the parallelogram $P$ is

\begin{pmatrix} &x_1 & x_2 \\ &y_1 & y_2 \end{pmatrix}

After change of variables the differential condition on $P$ can then be written

$(x_1y_2 + y_1x_2)f_{xy} + x_1x_2f_{xx} + y_1y_2f_{yy}$, I don't know that that is necessarily nonpositive.

Note that without the first bullet condition, the answer is no Planar function inequality on parallelograms.

I'm pretty sure the answer is yes with the additional condition - proof?

Answer 1

Take for instance $f(x,y)=x^2$, that satisfies the assumptions. Since one has $f(x,y)+f(1-x,1-y)= 1-2x+2x^2 < 1=f(0,0)+f(1,1)$ for all $0<x<1$, it verifies the stated inequality for no parallelogram (inscribed in $R$, with vertices in $(0,0)$ and $(1,1)$) with non-vertical edges.