Here is a simple proof using Theorem 1.6.1 in Lazarsfeld book, which is the following:
Theorem (Demailly)
If $H_1,\ldots,H_n$ are ample classes in a compact Kähler manifold of dimension $n$, then the following inequality holds:
$$(H_1 \cdots H_n)^n \ge (H_1^n)\cdots(H_n^n).$$
Let $H_1$ and $H_2$ be Kähler classes. By the AM–GM inequality,
$$\left[\frac{1}{2}\left( \frac{{H_1^{n-1}H_2}}{H_1^{n}} + \frac{H_2^{n-1}H_1}{H_2^{n}} \right)\right]^2 \ge\frac{{H_1^{n-1}H_2}}{H_1^{n}} \cdot \frac{H_2^{n-1}H_1}{H_2^{n}}. $$
Using the above theorem, it is easy to see that
$$({H_1^{n-1}H_2}) \cdot ({H_2^{n-1}H_1}) \ge {H_1^{n}}{H_2^{n}},$$
which finishes the proof.