A problem of Stanislaw Saks (Q2299453)

Let \(u\) be a subharmonic function on \(\mathbb{R}^{2}\) such that the partial derivatives \(u_{x},u_{y}\) exist on the coordinate axes, and \(u_{xx},u_{yy}\) exist at \((0,0)\). It is then shown that \(u_{xx}(0,0)+u_{yy}(0,0)\geq 0\). This gives an affirmative answer to Problem 184, posed by Saks in 1940, in the Scottish Book. The result does not extend to higher dimensions.