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PreprintCC BY 4.0Zenodo

A universal inequality index: U = 1 - e/P, P = ∏ 1/(1-g_n). From Γ(x)Γ(3-x) to the generating function of HHI, Shannon, Rényi, and Gini

  • Benito Vinuesa Guerrero
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Abstract

We introduce P(g)=∏(1−gn)−1 for partitions ∑gn=1 and the index U=1−e/P∈[0,1). P has universal infimum e at the infinitesimal equal partition and diverges at monopoly. Its log L=log⁡P=∑Sk/k, Sk=∑gnk, is the generating function of HHI, Simpson, R\'enyi, Shannon and Berger-Parker indices. U is additive, strictly Schur-convex, continuous and non-saturating, resolving infinite-tailed geometric vs harmonic decays where Gini collapses to 1. A finite-size normalization UN∗=1−Pmin/P recovers population invariance.

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Cite

BibTeXRIS
BibTeX
@misc{guerrero2026universal,
  title = {{A universal inequality index: U = 1 - e/P, P = ∏ 1/(1-g_n). From Γ(x)Γ(3-x) to the generating function of HHI, Shannon, Rényi, and Gini}},
  author = {Benito Vinuesa Guerrero},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23592342},
  url = {https://zenodo.org/records/23592342},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A universal inequality index: U = 1 - e/P, P = ∏ 1/(1-g_n). From Γ(x)Γ(3-x) to the generating function of HHI, Shannon, Rényi, and Gini
AU  - Benito Vinuesa Guerrero
PY  - 2026
DA  - 2026/10/23
PB  - Zenodo
DO  - 10.5281/zenodo.23592342
UR  - https://zenodo.org/records/23592342
AB  - We introduce P(g)=∏(1−gn)−1 for partitions ∑gn=1 and the index U=1−e/P∈[0,1). P has universal infimum e at the infinitesimal equal partition and diverges at monopoly. Its log L=log⁡P=∑Sk/k, Sk=∑gnk, is the generating function of HHI, Simpson, R\'enyi, Shannon and Berger-Parker indices. U is additive, strictly Schur-convex, continuous and non-saturating, resolving infinite-tailed geometric vs harmonic decays where Gini collapses to 1. A finite-size normalization UN∗=1−Pmin/P recovers population invariance.
ER  -