Entropy-like quantity

For $$p\in[0,1]^{\mathbb{N}}$$ and $$\alpha\ge1$$, define $$H_\alpha(p) = \sum_{i\in\mathbb{N}}p_i|\log(p_i)|^\alpha.$$ When $$\sum_i p_i=1$$ and $$\alpha=1$$, $$H_1(p)$$ is just the Shannon entropy of the distribution $$p$$. Has anyone encountered the object $$H_\alpha$$ anywhere in the literature? A reference would be much appreciated. I'm thinking of calling $$H_\alpha$$ hyperentropy for $$\alpha>1$$; is that term already taken by chance?

• Not an answer, but linking math.stackexchange.com/questions/3781318/… Sep 21 '20 at 23:19
• Thanks!! I'm curious about the other guy's motivation... Sep 21 '20 at 23:20

It's the $$\alpha^{\mathrm{th}}$$ moment of the Tribus surprisal.
So we ended up calling this quantity the $$\alpha$$th moment of information and proving some inequalities about it: https://arxiv.org/abs/2004.12680 (paper to appear in the NIPS 2021 conference).