Property 3.1. (Continuity). is a continuous function of P.
Property 3.2. (Symmetry). is a symmetric function of its arguments i.e.,
Property 3.3. (Normality). We have
Property 3.6. (Nonnegativity). with equality iff , where is a probability distribution such that one of the probabilities is one and all others are zero i.e.,.
Property 3.7. (Monotonicity). is a monotonic decreasing function of ( fixed).
Property 3.8. (Inequalities among generalized entropies). The following inequalities hold
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(3.12) |
where "" represents the logarithms with natural base "e".
Based on the expression (3.15), the properties 3.7 and 3.8 are extended as follows:
Property 3.7*. is also a decreasing function of ( fixed). In particular, when, the result still holds.
Property 3.8*. Parts (ii) and (iii) of the property 3.8 are simplified as follows:
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(3.13) |
Note 3.3. The condition is better than , , but it holds for . For , we need . In particular, when , the condition does not work. In this case, we need the second condition, i.e., , . In particular, when , or both the conditions, i.e., or , works well.
Property 3.11. (Concavity for n=2). For, is concave for all.
Property 3.12. (Schur-concavity). is a Schur-concave of .
Property 3.13. (Maximality). is maximum for the uniform probability distribution i.e.,
Property 3.15. (Quasiconcavity). is a quasiconcave function of P in.
Property 3.16. (Relative to maximum probability). Let , where, then the followings hold: