Geometric mean and harmonic mean

Inequality. $\sqrt{ab}\ge\frac{2ab}{a+b}$ is equivalent (since all quantities are positive) to
$$\sqrt{ab}(a+b)\ge2ab,$$
i.e. $a+b\ge2\sqrt{ab}$, which is exactly AM-GM.

Equality case. $a=b$.

Remark. In general, HM $\le$ GM $\le$ AM $\le$ QM (harmonic, geometric, arithmetic, quadratic means).