I am interested in a type of "mean" $r$ associated to a set $\{a_1,a_2,\dots,a_n\}$ where
$$ e^r=\frac{1}{n}\sum\limits_{i=1}^n e^{a_i}. $$
I will call this the "? mean" for now
The reason I believe this must have been studied is that if I instead consider $s$ for which
$$ {\rm ln}(s)=\frac{1}{n}\sum\limits_{i=1}^n {\rm ln}(a_i), $$ I get
$$ s=\big(\prod\limits_{i=1}^n a_i\big)^{1/n}; $$
in other words, the geometric mean. So to some extent we have
Geometric mean: Arithmetic mean :: Arithmetic mean : ? mean
Does anyone know what this kind of mean is called?
The geometric mean is the exponential of the arithmetic mean of the logarithms, i.e.
$$ \text{GM}(a_1,\ldots,a_n) = \exp\left(\text{AM}\left(\log a_1,\ldots,\log a_n\right)\right). $$ If you set
$$ \text{AM}(a_1,\ldots,a_n) = \exp\left(\text{UM}\left(\log a_1,\ldots,\log a_n\right)\right) $$
(where $\text{UM}$ stands for unknown mean) you get that $\text{UM}$ has the following property:
$$ \text{UM}(b_1,\ldots,b_n) = \log\text{AM}\left(e^{b_1},\ldots,e^{b_n}\right). $$
This is the mean in the log semiring, which is a smooth maximum.