The measure of uncertainty
given by (1.7) satisfy many interesting properties. For simplicity, we
shall take
or
instead of .
Unless otherwise specified, it is understood that
and .
Also it is understood that
and all the logarithms are with base 2.
Property 1.1. (Nonnegativity). with equality iff , where.
Property 1.2. (Continuity). is a continuous function of P.
Property 1.3. (Symmetry). is a symmetric function of its arguments, i.e.,
Property 1.4. (Expansible). We have
Property 1.5. (Decisive). We have
Property 1.6. (Normality). We have
Property 1.7. (Sum Representation). We can write
Property 1.8. (Recursivity). We have
Property 1.9. (Additivity). We have
Property 1.10. (Strongly Additive). We have
Property 1.12. (Generalized Grouping). We have
Property 1.13. (BinaryEntropic). Let

(1.8) 
Then

(1.9) 
with equality iff or iff (whenever for some , the corresponding is also zero).
Property 1.15. (Maximality). is maximum when all the probabilities are equal i.e.,
Property 1.16. (Uniform distribution). Let
Property 1.18. (Independence Inequality). If
Property 1.19. (Concavity). is a concave function of in .
Property 1.20. (Schurconcavity). is a Shurconcave function of in .
Property 1.21. Let be a probability distribution such that . Let us define such that Then .
Property 1.22. (Difference among two entropies). If
Property 1.23. Bounds on H(P). For , we have
Property 1.26. If is strongly additive (property 1.10), then it is additive (property 1.9).
Property 1.27. If is additive (property 1.9) for and is expansible (property 1.4), then it is also additive (property 1.9).
Property 1.28. If is expansible (property 1.4) and strongly additive (property 1.10), then it is recursive (property 1.8).
Property 1.29. If is recursive (property 1.8) for and symmetric (property 1.3) for , then it is symmetric (property 1.3) for and decisive (property 1.5).
Property 1.30. If is recursive (property 1.8), symmetric (property 1.3) for , then it is symmetric (property 1.3) and expansible (property 1.4).
Property 1.31. If is symmetric (property 1.3) and recursive (property 1.8), then
Property 1.33. If is expansible (property 1.4) and subadditive (property 1.17) for , then it is nonnegative (property 1.1).
Property 1.34. If is branching of the form
Property 1.35. If is recursive (property 1.8), and , then
Property 1.36. If is normalized (property 1.6), symmetric (for ) (property 1.3) and recursive (for ) (property 1.8), then the function satisfies the property 1.13(i)(iv).
Property 1.37. Binary entropic properties given by 1.13(i)(iv) implies that is symmetric (property 1.3), normalized (property 1.6), expansible (property 1.4), decisive (property 1.5), recursive (property 1.8), strongly additive (property 1.9) and additive (property 1.10).
Note 1.1. Some of the properties given above can be seen in
Aczél and Daróczy (1975) [2]
and Mathai and Rathie (1975) [71]