A monotonicity property of variances

We prove that variances of non-negative random variables have the following monotonicity property: For all $0<r<s\le 1$, and all $0\le X\in L^2$, we have $\mathrm{Var}(X^r{)}^{1\mathrm{/}r}\le \mathrm{Var}(X^s{)}^{1\mathrm{/}s}$. We also discuss the real valued case.