Entropies and Their Concavity and Schur-Concavity Conditions

Abstract

Concavity and Schur-concavity are two of the important properties of any entropy. Since Shannon’s classical entropy formulation, a number of generalized entropies have been proposed as parameterized generalizations of Shannon’s entropy. For such generalized entropies, the conditions under which they are concave and/or Schur-concave have not always been determined or have been incompletely and incorrectly reported in a variety of publications. This paper provides proofs of those two properties for the various proposed generalized entropies using a unifying approach. First, a new three-parameter entropy is introduced of which other proposed generalized entropies are particular members. Second, a proof is derived for the concavity and Schur-concavity of the new entropy and the underlying conditions. Those results are then applied to the particular one-parameter and two-parameter members. Some new such members are also discussed as are some related inequalities. The various derivations are based on so-called generalized probability distributions when the sum of component probabilities may be less than 1.

Keywords

• Entropy
• concavity
• Schur-concavity
• generalized entropies
• Gini’s means