Arithmetic-Harmonic Geometric-Harmonic or mean?

Question

There is a Arithmetic-Geometric Mean that appears to be common and very useful throughout several branches of math. Is there a Arithmetic-Harmonic Mean and a Geometric-Harmonic mean? What are their properties and are the useful? Is there a general f(x)-g(x) mean? What properties would that have.

Answer 1

The AHM is just the geometric mean. This is easy to see, after proving they have a common limit: $$ a_{n+1} = \frac{1}{2}(a_n+b_n), \quad b_{n+1} = \frac{2a_nb_n}{a_n+b_n} = \frac{a_n b_n}{a_{n+1}}, $$ so $ a_{n+1}b_{n+1}=a_nb_n = \dotsb = a_0 b_0 $; the common limit is then $a_{\infty}^2=a_0 b_0$, so $a_{\infty}=b_{\infty}= \sqrt{a_0b_0} $.

Meanwhile, the HGM is simply related to the AGM: $HGM(a,b)=1/AGM(1/a,1/b)$ (a proof is found in this answer).

As for generalisations, there are loads. See for example this MathOverflow post, and this writeup about the generalisation to some sort of algebraic geometry, with plenty of nice $\theta$-functions. Indeed, I would have thought that any iteration of a homogeneous function on the positive quadrant of $\mathbb{R}^2$ which is a contraction with respect to $\lvert x-y\rvert$ will give an analogous merged mean.