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Unified (r,s)-Entropy

Instead of studying the properties of the entropies (3.1), (3.3), (3.5), (3.6) and (3.7) separately, our aim here is to study them in a unified way. This unification is as follows:

$${\bf {\ensuremath{\boldsymbol{\mathscr{H}}}}}^s_r(P)= \left\{ \be......neq 1 \\ H^1_r(P), & r\neq 1, \, s=1 \\ H(P), & r=1, \, s=1\end{array}\right.$$ (3.8)

for all $r \in (0,\infty)$ and $s \in (-\infty,\infty)$, where$H^1_r(P)$ is the same as $H_r(P)$ given in (3.1). From now onwards, we shall use the notation $H^1_r(P)$ instead of $H_r(P)$.

The entropies $H^s(P)$ and $_tH(P)$ do not appear in the unified expression (3.8) because they are particular cases of$H^s_r(P)$, and hence are already contained in it. According to the notations above, $H^s(P)$ given in (3.3) means $H^s_s(P)$ for$r=s$ in $H^s_r(P)$. Henceforth, we shall consider this notation too.

It is customary to study the generalized measure given in (3.8) for positive values of $r$ and $s$. Taneja (1989) [105] and Taneja et al. (1989) [109] studied then for $r>0$ and any $s$. The natural question arises, why to keep$r>0$? Why not to study them for $r\leq 0$. This is our aim in this chapter i.e., to study the measure (3.8) for $r,s\\varepsilon \ (-\infty, \infty)$. By considering $r<0$, again the problem arises is that, we can find probability distributions such that the measure (3.8) become infinite. To avoid this, we shall redefine the set $\Delta_n$ in the following way: 

$$_r\Delta_n=\Big\{P=(p_1,...,p_n)\Big\vert \sum_{i=1}^{n}p_i=1,\,\,$$with$$\,\, p_i\geq 0,\,\, r>0\,\,$$and$$\,\,p_i>0,\,\, r\leq 0, \,\,\forall i\Big\}.$$ Unless otherwise specified from now onwards, we shall consider the measures ${\bf {\ensuremath{\boldsymbol{\mathscr{H}}}}}^s_r(P)$, for all $P, Q \in$$_r\Delta_n,\, r,s \in (-\infty,\infty)$.

We call, the unified expression (3.8), i.e.,${\bf {\ensuremath{\boldsymbol{\mathscr{H}}}}}^s_r(P)$, the unified $(r,s)-$entropy.

From the unified expression (3.8), we observe that${\bf {\ensuremath{\boldsymbol{\mathscr{H}}}}}^s_r(P)$ is a continuous extension of $H^s_r(P)$ with respect to the parameters. In order to study the properties of ${\bf {\ensuremath{\boldsymbol{\mathscr{H}}}}}^s_r(P)$ it is sufficient to study the properties of $H^s_r(P)$ for $r\neq 1, \ s\neq 1$, because the rest part follows by continuity with respect to $r$ and $s$. For simplicity, this observation is denoted as follows:

$${\bf {\ensuremath{\boldsymbol{\mathscr{H}}}}}^s_r(P)=CE\Big\{ H^s_r(P):r\neq 1,\ s\neq 1\Big\},$$ (3.9)

where "CE" means "continuous extension" with respect to the parameters $r$ and $s$.

Note 3.1. The notations ${\bf {\ensuremath{\boldsymbol{\mathscr{H}}}}}^1_r(P)$ and ${\bf {\ensuremath{\boldsymbol{\mathscr{H}}}}}^s_1(P)$ are understood as follows:

$${\bf\ensuremath{\boldsymbol{\mathscr{H}}}}^1_r(P)=\left\{\begin{array}{ll}H^1_r(P), & r\neq 1 \\ H(P), & r=1\end{array}\right.$$ and $${\bf {\ensuremath{\boldsymbol{\mathscr{H}}}}}^s_1(P)=\left\{\begin{array}{ll}H^s_1(P), & s\neq 1 \\ H(P), & s=1\end{array}\right.$$ respectively, for all $P=(p_1,p_2,...,p_n)\ \in\ \Delta_n$, $r \in (0,\infty)$ and $s \in (-\infty,\infty)$.

Subsections

• Composition Relations Among the Unified (r,s)-Entropy Measures
• Properties of Unified (r,s0-Entropy