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List of Generalized Entropies

For all $P=(p_1,p_2,...,p_n)\ \in\ \Delta_n$ the following generalized entropies are known in the literature by their respective authors, starting with Shannon's (1948) [86]. In some cases, it is understood that $p_{i} > 0$ for all$i=1,2,...,n$. By no means can we say that the list is complete. Graphic representation of these entropies can be seen in Taneja (1989; 1990a) [105], [106], where it is shown that how some of these measures approaches to Shannon's entropy.

Shannon (1948) [86]

$\Phi_1(P)=-\sum_{i=1}^n{p_i\log \, p_i}.$

Rényi(1961) [82]

$\Phi_2(P)=(1-r)^{-1}\log\Big(\sum_{i=1}^n{p^r_i}\Big),\ r\neq 1,\ r>0.$

Aczél and Daróczy (1963) [1]

$\Phi_3(P)=-\sum_{i=1}^n{p^r_i\log \,p_i}/\sum_{i=1}^n{p^r_i},\ r>0$

$\Phi_4(P)=(s-r)^{1}\log\Big(\sum_{i=1}^n{p^r_i}/\sum_{i=1}^n{p_i^s}\Big), \ r\neq s,\r>0,\ s>0$

$\Phi_5(P)={1\over s}\arctan \Big\{\sum_{i=1}^n{p^r_i \sin(s\log p_i)}/\sum_{i=1}^n{p^r_i \cos(s\logp_i)}\Big\},$

$$s\neq 1, s>0, r>0.$$

Varma(1966) [119]

$\Phi_6(P)={1\overm-r}\log\Big(\sum_{i=1}^n{p^{r-m+1}_i}\Big), \ m-1<r<m,\ m\geq 1.$

$\Phi_7(P)={1\overm(m-r)}\log\Big(\sum_{i=1}^n{p^{r/m}_i}\Big), \ 0<r<m,\ m\geq 1.$

Kapur (1967) [54]

$\Phi_8(P)=(1-t)^{-1}\log\Big(\sum_{i=1}^n{p^{t+s-1}_i}/\sum_{i=1}^n{p^s_i}\Big),\ t\neq 1,\ t>0,\ s\geq 1.$

Havrda and Charvát (1967) [46]

$\Phi_9(P)=(2^{1-s}-1)^{-1}\Big[\sum_{i=1}^n{p^s_i-1}\Big], \s\neq 1,\ s>0.$

Belis and Guiasu (1968) [10]

$\Phi_{10}(P)=-\sum_{i=1}^n{p_i w_i \log p_i},\w_i>0,\ i=1,2,...,n.$

Rathie (1970) [78]

$\Phi_{11}(P)=(1-r)^{-1}\log\Big(\sum_{i=1}^n{p^{r+s_i-1}_i}/\sum_{i=1}^n{p^{s_i}_i}\Big),$

$$s_i\geq 0,\ i=1,...,n,\ r\neq 1,\ r>0.$$

Arimoto (1971) [3]

$\Phi_{12}(P)=(2^{t-1}-1)^{-1}\Big[\Big(\sum_{i=1}^n{p^{1/t}_i}\Big)^t-1\Big],\ t\neq 1,\ t>0.$

Sharma and Mittal (1975) [90]

$\Phi_{13}(P)=(2^{1-s}-1)^{-1}\Big[\exp_2\Big((s-1)\sum_{i=1}^n{p_i\logp_i}\Big)-1\Big], \ s\neq 1,\ s>0.$

$\Phi_{14}(P)=(2^{1-s}-1)^{-1}\Big[\Big(\sum_{i=1}^n{p^r_i}\Big)^{s-1\overr-1}-1\Big], \ r\neq 1,\ s\neq 1,\ r>0,\ s>0.$

Sharma and Taneja (1975; 1977) [92] [93]

$\Phi_{15}(P)=-2^{r-1}\ \sum_{i=1}^n{p^r_i\logp_i}, \ r>0$

$\Phi_{16}(P)=(2^{1-r}-2^{1-s})^{-1}\\sum_{i=1}^n{p^r_i-p^s_i}, \ r\neq s,\ r>0,\ s>0$

$\Phi_{17}(P)=-{2^{1-r}\over \sin s}\\sum_{i=1}^n{p^r_i\ \sin(\mbox{ s\,log}\ p_i)}, \ s\neq k\pi,\k=0,1,..., \ r>0$

Picard (1979) [77]

$\Phi_{18}(P)=-\sum_{i=1}^n{v_i\logp_i}/\sum_{i=1}^n{v_i}$

$\Phi_{19}(P)=(1-r)^{-1}\log\Big(\sum_{i=1}^n{p^{r-1}_iv_i}/\sum_{i=1}^n{v_i}\Big), \ r\neq 1,\ r>0$

$\Phi_{20}(P)=(2^{1-s}-1)^{-1}\Big[\exp_2\Big((s-1)\sum_{i=1}^n{v_i\logp_i}/ \sum_{i=1}^n{v_i}\Big), \ s\neq 1,\ s>0$

$\Phi_{21}(P)=(2^{1-s}-1)^{-1}\Big[\Big(\sum_{i=1}^n{p^{r-1}_iv_i}/\sum_{i=1}^n{v_i}\Big)^{s-1\over r-1}-1\Big],$

$$r\neq 1,\ s\neq1,\ r>0,\ s>0$$

Ferrari (1980) [37]

$\Phi_{22}(P)=(1+{1\over \lambda}\log(1+\lambda)-{1\over \lambda} \sum_{i=1}^n{(1+\lambdap_i)\log(1+\lambda p_i)},\ \lambda >0.$

Sant'anna and Taneja (1985) [85]

$\Phi_{23}(P)=-\sum_{i=1}^n{p_i\log \big(\sinsp_i/2\sin {s\over 2}\big)},\ 0<s<\pi$

$\Phi_{24}(P)=-\sum_{i=1}^n{\Big({\sin sp_i\over2\sin (s/2)}\Big)\log \Big({\sin sp_i\over 2\sin(s/2)}\Big)},0<s<\pi$

$\Phi_{25}(P)=\sum_{i=1}^n{{\sin sp_i\over 2\sin(s/2)}},0<s<\pi$

Kapur (1988) [59]

$\Phi_{26}(P)=-\sum_{i=1}^n{\log \Gamma(1+p_i)},$

where $\Gamma$ is the well known gamma function.