Let \(\Omega\) be an open set in \(\mathbb{R}^ m \times \mathbb{R}^ n\). A function \(u\) is separately subharmonic on \(\Omega\) if \(u(x,\cdot)\) is subharmonic on the \(x\)-section of \(\Omega\), and \(u(\cdot,y)\) on the \(y\)-section, for all \((x,y) \in\Omega\). The main result states that, if \(m \geq n \geq 2\), \(u\) is separately subharmonic on \(\Omega\), and there is a nonnegative increasing function \(\varphi\) on \([0,\infty[\) such that \[ \int^ \infty_ 1 s^{(n-1)/(m-1)} \varphi(s)^{1/(1-m)} ds<\infty \] and \(\varphi (\log^ +u^ +)\) is locally integrable on \(\Omega\), then \(u\) is subharmonic. The proof uses an argument of \textit{Y. Domar} [Ark. Mat. 3, 429-440 (1958; Zbl 0078.093)]. An example shows that the above result is almost sharp.