A sufficient condition for planar partition functions (Q1898874)

A twice differentiable function \(\mathbb{R}\to\mathbb{R}\) that has a positive (negative) second derivative is convex (concave). If it satisfies one more condition, it is a planar partition function. In this note we derive and discuss a sufficient condition for a twice continuously differentiable function \(f:\mathbb{R}^2\to\mathbb{R}^2\) to be a planar partition function. To this end we define a ``second derivative'' \(f''\) of \(f\) which is a continuous function \(\mathbb{R}^2\to\mathbb{R}\). The fact that this second derivative is negative can be used to conclude that \(f\) is a planar partition function in very much the same way as a (usual) negative second derivative for a function \(g:\mathbb{R}\to\mathbb{R}\) can be used to conclude that \(g\) is a planar partition function. A negative second derivative at a point also seems to imply that the function is, in a certain sense, convex in a neighborhood of this point. We will discuss this observation in a final section. Also included in this note is a list of the known continuous planar functions \(\mathbb{R}^2\to\mathbb{R}^2\). We use the examples in this list to test our condition.