Let \(\tau\) be the fundamental superharmonic function of \({\mathbb{R}}^ N\) at pole 0, let D be a proper subdomain of \({\mathbb{R}}^ N\) and d be the distance function from \(\partial D\). The author proves that if \(N=2\) then the function \(\tau\) (d) (which is subharmonic in D) has a harmonic majorant \({\hat \tau}\)(d) in D if and only if \(\partial D\) consists of only one point while, in the case where \(N>2\), \({\hat \tau}\)(d) exists if and only if \(\partial D\) consists of a countable number of isolated points \(\{x_ n\}\) and \(\sum_{x_ n\neq 0}\tau (\| x_ n\|)<\infty\). Another result is that if D is any domain in \({\mathbb{R}}^ N\) and \(\rho\) is a positive function on D such that \(\overline{B(x,\rho (x))}\subset D\) for all x in D then, for all u superharmonic in D, the \(\rho\)-mean function \(u_{\rho}\) of u in D \((u_{\rho}(x)=M(u,x,\rho (x))\) is superharmonic in D if and only if \(\tau\) (\(\rho)\) is superhamonic in D. When \(D\neq {\mathbb{R}}^ N\) such a \(\rho\) exists if and only if D is one of the domains described in the first result mentioned above.