And did anyone study the sequence of means introduced here?

Many. Check the Genaralized Mean

$$M_p(x_1,\dots,x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{\frac{1}{p}} $$

in

https://en.wikipedia.org/wiki/Generalized_mean

And did anyone study the sequence of means introduced here?

Many. Check the Generalized Mean

$$M_p(x_1,\dots,x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{\frac{1}{p}} $$

in

https://en.wikipedia.org/wiki/Generalized_mean