Power means are defined by $$ \left(\frac{1}{n}\sum_i x_i^p\right)^{1/p} $$
where $p\in\mathbb{R}$, and the $x_i$ are positive real numbers.
I wonder what happens when some of the $x_i$ are negative. Then either the sum $s$ may be negative, in which case $s^{1/p}$ may be a complex number or $x_i^p$ is directly a complex number. I have two questions on this:
Is taking the power means in these cases still a well-defined operation? Does it have a semantics by analogy to the case when the $x_i$ are positive?
What could we do to remedy this? Adding a shift to all numbers, taking the power mean and then shifting back would be a solution. But does that make any sense?
Are there other forms of generalized means applicable to positive and negative reals?