Let denote the geometric mean of the binomial coefficients

Prove that .

**Solution**

We note that

On the other hand the following lemma holds:

**Lemma: **Let be a monotonic function. It holds that

*Proof: Due to monotony it holds that **for . Hence summing over all these values of k we get that*

*The result follows.*

Applying the above to on we get that:

Thus,

(1)

Similarly, applying the above to on we get that:

(2)

The result follows.