This might seem a bit left-field, but it came up rather naturally in some work I've been doing recently in machine learning and I can't seem to track down any work along the same lines. In essence, if you start with a (convergent) power series $f:\mathbb{R}_+ \to \mathbb{R}_+$ with non-negative weights $a_i$: $$f(x) = \sum_{i=1}^\infty a_i x^i$$ you can construct a "series-like" function $g:\ell^1(\mathbb{N}) \to \mathbb{R}_+$: $$g(v) = \sum_{i=1}^\infty a_i \|v\|_i^i$$ in terms of the $p$-norms $\|\cdot\|_p$.
This seems reasonably well-behaved ($g(v) \leq f(\|v\|_1)$, and $f=g$ if $n=1$ for example, and I believe $g$ defines an F-space) but I wonder if this has been studied before? In addition to any interesting/novel properties (which I am of course interested in), I'm particularly wondering if $g(v)$ acts like $h(\|v\|)$ for some norm $\|\cdot\|$ and function $h$ (such a property would be useful, but not essential, for my particular application).
Sorry if that's a broad question but this is one of those cases where either there isn't much prior work or (more likely) I just don't know the keywords I need to google.