Go to:Table of Contents

Logic Among Eight Entropies

In this chapter, we have presented seven generalized entropies. Four of them are with one parameters and three are with two parameters. Including Shannon's entropy, in all, we have eight entropies. Now we shall present a logic i.e., relationship among these eight entropies.

We can write

1. $H^s_r(P)=M_s(H^1_r(P)).$

2. $H^s_1(P)=M_s(H(P)).$

3. $H^s_s(P)$ as a particular case of $H^s_r(P)$ for $r=s$.

4. $_tH(P)$ as a particular case of $H^s_r(P)$ for $t=r^{-1}=2-s$.

5. $H_{r,s}(P)=\lambda_1 H^1_r(P)+(1-\lambda_1)H^1_s(P)$, where $\lambda_1 ={1-r\over s-r},\,\, s\neq r$.

6. $H^{r,s}(P)=\lambda_2 H^r_r(P)+(1-\lambda_2)H^s_s(P)$, where $\lambda_2 ={2^{1-s}-1\over 2^{1-r}-2^{1-s}}, s\neq r$.

Thus, the entropies $H^1_r(P)$ and $H(P)$ are the composite forms of the entropies $H^s_r(P)$ and $H^s_1(P)$ respectively. The entropies $H^s_s(P)$ and $_tH(P)$ are particular cases of$H^s_r(P)$. The entropies $H_{r,s}(P)$ and $H^{r,s}(P)$ bear linear relations with $H^1_r(P)$ and $H^s_s(P)$ respectively.

The above logic have been adopted by Capocelli and Taneja (1985) [23] to study the properties of the above eight entropies. This logic works well specially towards inequality properties. The another approach of unification by using continuity property with respect to the parameters, have been applied by Taneja (1989) [105].