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Entropies of Order 1 and Degree s and Order r and Degree s

Sharma and Mittal (1975) [90] introduced and characterized two entropies called entropy of order $1$ and degree $s$ and entropy of order $r$ and degree $s$ given by

$$H^s_1(P)={(2^{1-s}-1)}^{-1}\bigg[\exp_2\bigg((s-1)\sum_{i=1}^n{p_i\log\p_i}\bigg)-1\bigg],\ s\neq 1,$$ (3.6)

and

$$H^s_r(P)={(2^{1-s}-1)}^{-1}\bigg[{\bigg({\sum_{i=1}^n{p^r_i}}\bigg)}^{s-1\overr-1}-1\bigg],\ r\neq 1, s\neq 1, r>0,$$ (3.7)

respectively, for all $P=(p_1,p_2,...,p_n)\ \in\ \Delta_n$, where$r$ and $s$ are real parameters.

Sharma and Mittal's main motivation was to generalize the three entropies, $H_r(P), \,\,H^s(P),$ and $_tH(P)$. With this aim, they arrived at $H^s_r(P)$. $H^s_r(P)$ reduces to $H^s(P)$ and $_tH(P)$, when $r=s$ and $r^{-1}=t=2-s$, respectively.$H^s_r(P)$ reduces to $H^s_1(P)$ and $H_r(P)$ when $r\to 1$ and$s\to 1$ respectively. Also, $H^s_1(P)$ reduces to Shannon's entropy, H(P), when $s\to 1$.

Thus, we see that the entropy of order $r$ and degree $s$ contain, either as a limiting or as a particular case, the Shannon's entropy, the entropy of order $r$, the entropy of degree $s$, the entropy of kind t, and the entropy of order 1 and degree $s$.

We have presented in the previous subsections seven generalized entropies having one and/or two scalar parameters. A natural question arises at this stage, what kind of other generalized entropies exist in the literature. For a complete list we refer to section 3.6.2 (Taneja, 1989; 1990a) [105], [106].