Power mean $M_p(a,b)$ of order $p \in \mathbb{R}$ for a pair $(a,b) \in \mathbb{R}^+$ is defined as $M_p(a,b)= \Big(\frac{a^p+b^p}{2}\Big)^{\frac{1}{p}}$. For example $p = 1$ gives arithmetic mean, also $p = -1$, $p = 2$ and $p \rightarrow 0$ give harmonic, quadratic and geometric mean.
But we also have $p \rightarrow +\infty$ that gives $M_{+\infty}(a,b) = \max(a,b)$ and $p \rightarrow -\infty$ gives $M_{-\infty}(a,b) = \min(a,b)$, which is quiet clear as $a$ or $b$ will dominate the other. If we try to generalise for $(a,b) \in \mathbb{R}^2$, we get the problem that for $a < 0$ and $b < 0$ we can only take $p$ as an integer, and moreover $M_p$ with $p \rightarrow \infty$ doesn't have a limit since $p$ even gives positive limit and $p$ odd gives negative limit, we can avoid this problem with defining the limit only with $p$ odd. Note also that the conventional definitions of $\max$ and $\min$ are not respected anymore with $a < 0$ and $b < 0$ but it's not a problem since we only search to have a "smooth" generalisation.
There's some questions that come to mind on complex generalisation, the question 2 is the main that I would like to know but the others are secondary:
- For which $(a,b) \in \mathbb{C}^2$ do these two limits exist using $p$ an odd integer? If it doesn't exist, can we still make it work restricting some cases?
- When it's well-defined for a large portion of $(a,b) \in \mathbb{C}^2$, $\underline{\text{what exactly do the numbers}}$ $M_{\pm \infty}(a,b)$ $\underline{\text{correspond to geometrically in the complex plane in relation to affix points}}$ $a$ $\underline{\text{and}}$ $b$?
- When does $M_{\pm \infty}(a,b)$ map to either $a$ or $b$?
- Is there a link between this continuation, and the continuation for $\max(a,b)$ that would return $a$ if $|a| > |b|$ and return $b$ if $|b| > |a|$? (we only study the case where $|a| \neq |b|$ here, it doesn't make sense otherwise)
I didn't find any other post on maths exchange that talks about this (aside from this one, or numerous ones asking if complex can be ordered), also sorry if I missed it, I also didn't check mathoverflow. Thanks in advance if you have any ideas! :)