# Geometric mean "flipped" at arithmetic mean?

Due to the inequality $$2\sqrt{xy}\leq x+y$$, the geometric mean is always closer to the smaller value than the arithmetic mean.

In my situation, I need a "mean" that is closer to the larger value, so I thought of simply "flipping" the geometric mean $$g(x,y)$$ at the arithmetic mean $$a(x,y)$$: $$a(x,y) + \Big(a(x,y)-g(x,y)\Big) = x+y-\sqrt{xy}$$ The result is always between the arithmetic mean and the larger value.

Is there some special name for this "mean"?

• If you want a mean that emphasizes larger values, you could consider a a root-mean-square, or more generally $A(x^p)^{1/p}$, where $A()$ is the arithmetic mean, for $p > 1$ ... (RMS corresponds to $p=2$) Apr 12, 2023 at 15:11
• Ordinarily, to justify calling a function $f(\ ,\ )$ a "mean," you would like it to be increasing in both variables. With $f(x,y)=x+y-\sqrt{xy}$ (defined on non-negative values only), the partial derivative in the first variable is $1-\sqrt{y/(4x)},$ which is negative when $y \gt 4x.$ That's not a mean. Instead of guessing formulas and checking their properties, it would be more appropriate to develop a mean based on the characteristics of your underlying problem: what exactly is this mean intended to represent?
– whuber
Apr 12, 2023 at 15:23
• @BenBolker Wikipedia calls it generalized mean, power mean or Hölder mean. Apr 12, 2023 at 15:57
• I'm not sure if I should answer or @whuber should - both comments seem valuable. It would be fine with me if you (cdalitz) composed an answer to your own question that incorporates whatever information from the comments is useful. Apr 12, 2023 at 16:50
• You can describe the geometric mean as $\exp\left (\frac1n \sum \log_e(x_i)\right)$ so one possible flipped version might be $\log_e\left (\frac1n \sum \exp(x_i)\right)$. This will be closer to the maximum value but will not exceed it. A curious effect is that while the geometric mean is in a sense scale-invariant but not location-invariant, this flipped version would in a similar sense be location-invariant but not scale-invariant (the arithmetic mean is both). Apr 12, 2023 at 17:16

## 1 Answer

Thanks for the valuable comments, which I try to summarize and put into a unifying framework in this answer.

As pointed out by @whuber, my suggested formula violated one plausible axiom for a mean, namely that it should increase in all arguments. @whuber also suggested to base the formula on a more rigorous axiomatic ground by postulating a number of reasonable properties of the desired "mean".

Hence I did a brief literature study, and it turned out that none less than Kolmogoroff (sic!) already did exactly that. In 1930, he postulated the following properties for a function $$M:{\mathbb R}^n \to{\mathbb R}$$ that represents a "regular mean":

1. $$M$$ is continuous and increasing in each variable.
2. $$M$$ is a symmetric function.
3. The mean of repeated data equals the repeated value.
4. The mean of a sample remains unchanged if a part of the sample is replaced by its corresponding mean

Kolmogoroff proved that, if these conditions hold, the mean must be of the form $$M(x_1,\ldots,x_n) = f^{-1}\left(\frac{1}{n}\sum_{i=1}^n f(x_i)\right)$$ which the Wikipedia page cited by @COOLSerdash calls "generalized f-mean", and the French Wikipedia calls it "quasi-arithmetic mean" or "Kolmogoroff mean".

A. Kolmogoroff: "Sur la notion de la moyenne." Atti Reale Accademia Nazionale dei Lincei, vol. 12,‎ 1930, p. 388–391

When applying this to my particular problem of defining a mean that is greater than the arithmetic mean, it is sufficient that $$f$$ is a convex function, because for a convex function we have $$\frac{1}{2}\Big(f(x)+f(y)\Big) \geq f\left(\frac{x+y}{2}\right) \Rightarrow f^{-1}\left(\frac{1}{2}(f(x)+f(y))\right) \geq \frac{x+y}{2}$$ The Hölder mean of order $$p$$, which is the root-mean-square for $$p=2$$, as suggested by @BenBolker, is the special case $$f(x)=x^p$$. The choice $$f(x)=e^x$$, as suggested by @Henry, is yet another special case.

For my use case, I have settled on the Hölder mean of order two.

• +1 nice, authoritative summary.
– whuber
Apr 13, 2023 at 13:48