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 Kähler 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.

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 Kähler 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.

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.

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 Kähler 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.