= Solution
By <asymptotic Riemann–Roch> and the preceding bounds for higher <sheaf cohomology>,
$$
\boxed{h^0(X,mD)-h^1(X,mD)=\frac{D^n}{n!}m^n+O(m^{n-1}).}
$$
A <nef divisor> satisfies $D^n\geq0$: for a fixed <ample divisor> $A$, the divisors $D+\varepsilon A$ are ample for positive rational $\varepsilon$, and continuity of the <intersection product> gives the inequality as $\varepsilon\to0$. If $h^0(X,mD)=0$ eventually, the left side is nonpositive. Dividing by $m^n$ forces $D^n=0$, since $h^1\geq0$ and $D^n\geq0$. The same identity then gives $h^1(X,mD)=O(m^{n-1})$.
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