It is proved the following: If $u$ is a function harmonic in the unit ball $B\subset\mathbb R^N$, and $0<p<1$, then there holds the inequality \[ \underset{0<r<1}\sup\int_{\partial B}|u(ry)|^pd\sigma\leq|u(0)|^p+C_{p,N}\int_B(1-|x|)^{p-1}|\nabla u(x)|^pdV(x). \] In the case $p>(N-2)/(N-1)$, this was proved by Stevi\'c [17].