For fixed \((p,q)\in \mathbb R^2\) the author characterizes Schur \(m\)-power convex and concave Stolarsky means \[ S_{p,q}(a,b):= \left(\frac{q(a^p-b^p)} {p(a^q-b^q)}\right)^{\frac{1}{p-q}}\quad(pq(p-q)\neq 0, a\neq b,a,b>0) \] with respect to the variables \((a,b)\in \mathbb R^2\). E.g. for \(m>0\) the mean \(S_{p,q}\) Schur \(m\)-power is convex (concave) if and only if \[ \min\left\{\frac{p+q}{3},\, \min\{p,q\}\right\}\geq m \,(\leq m). \] The proofs follow standard arguments with some elaborate calculations.