I learned here that there is a relation between weighted means of the form $x_1^{\lambda_1}\dotsb x_n^{\lambda_n}$ and $(\lambda_1 x_1^r + \dotsb + \lambda_nx_n^r)^{1/r}$, namely that the former is the limit of the latter as $r\to 0$ (so the geometric mean is the $0$th power mean).
I learned here that there is an inequality $x_1^{\alpha_1} + \dotsb + x_n^{\alpha_n}\geq x_1^{h/n}\dotsb x_n^{h/n}$, where $h = h(\alpha_1, \dotsc, \alpha_n)$ is the harmonic mean of the $\alpha_i$, where $x_i>0$ and $\alpha_i >0$.
My question is: are the two related? I was looking for a way to generalize the second result above, and I wondered if it was naturally related to generalized power means. Any ideas, folks?