Recently, while playing around with infinite-divisibility, i arrived at the following metric:

$$d(x,y) := \sqrt{\log\left(\frac{x+y}{2\sqrt{xy}}\right)},$$

defined for positive reals $x$ and $y$. Proving that $d$ is a metric is trivial, except for the triangle-inequality. However, we can bypass a direct proof by appealing to Schoenberg's theorem (I. J. Schoenberg. *Metric spaces and positive-definite functions*, TAMS, 1938), from which the metricity follows easily because $-\log(x+y)$ is a conditionally positive-definite kernel.

However, i have been searching for following:

- Applications / situations where this metric shows up?
- An elementary proof of $d(x,y)$ being a metric.

**Remarks**

a. A google search on "ratio arithmetic geometric mean" yields some applications of the *ratio* alone;

b. An elementary proof should exist, but my initial attempts have not been that successful, especially as i stubbornly did not want to use differential calculus.

c. Notice that while proving $$d(x,y) \le d(x,z) + d(y,z),$$ we may assume wlog $x < 1$ and $y > 1$ and $z=1$, as proving the other cases ranges from very-trivial to trivial.