Schur-Concavity of Rényi Heterogeneity

Abraham Nunes MD PhD MBA

Dalhousie University, Halifax, Nova Scotia, Canada

The Rényi heterogeneity of system *X* defined on state space X={1,2,..., n}, with probability distribution is

The Schur-Ostrowski criterion states that if *f* is a symmetric function and all first partial derivatives exist, then *f* is Schur-concave iff

holds for all *1**≤**i**≠**j**<**n*

PROPOSITION 1 (Symmetry of Rényi Heterogeneity). Given a probability vector and permutation function , the Rényi heterogeneity

,

satisfies

.

The proof trivially follows from the commutativity of addition.

PROPOSITION 2 (Differentiability). Given a probability vector and the Rényi heterogeneity

,

we have that ∃ .

Proof. At q∉{0,1,∞}, we have

At *q**=**0*, we have

At *q**=**1* we have

At *q**=**∞* we have

THEOREM (Schur-Concavity of Rényi Heterogeneity). The Rényi heterogeneity is Schur-concave.

Proof.