# Class of functions with negative mixed partial derivatives

Lets say we define a class of functions $g: \mathbf{R}^2 \rightarrow \mathbf{R}$ by the requirement that

$$\frac{\partial^2 g}{\partial x_1 \partial x_2}(x_1,x_2) \le 0$$

for all $x_1$ and $x_2$. What is the name of this class?

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Hm ... one could call it the class of functions with negative mixed second partial derivatives ... I don't think this class has a special name. – martini Jun 27 '12 at 19:29

This is an interesting class of functions with no established name. The definition has an appealing derivative-free reformulation which allows the class to include nonsmooth functions at all. $$(*)\qquad g(u_2,v_2)+g(u_1,v_1)\le g(u_2,v_1)+g(u_1,v_2),\quad \text{whenever } u_2\ge u_1, v_2\ge v_1$$ For smooth functions (*) is equivalent to the mixed-partial inequality.
The form of (*) suggests some sort of rearrangement inequality: the sum gets smaller when the sequences $u_i$ and $v_i$ are arranged in nondecreasing way. And indeed, this class appears (without a name) in the paper Symmetric decreasing rearrangement is sometimes continuous by Almgren and Lieb, see Theorem 2.2. Actually, Almgren and Lieb work with the reverse inequality, but this is a minor point ($g$ vs $-g$). My PhD advisor called the functions $F$ with $F(u_2,v_2)+F(u_1,v_1)\ge F(u_2,v_1)+F(u_1,v_2)$ "AL functions" in honor of Almgren and Lieb, but never in print as far as I know.