We give sufficient conditions for a nonnegative function to behave like a subharmonic function. If $f$ is a $C^1$-function on a domain $D\subset R^n$ such that $|\nabla f(a)|\leq Kr^{-1}$ $\omega_f(a,r)$ ($K=$const) where $\omega_f(a,r)$ is the oscillation of $f$ on the ball $B_r(a)\subset D$, then both $|f|^p$ and $|\nabla f|^p$ ($p>0$) have a weakened sub-mean-value property.