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Entropy of Degree (r,s)

Sharma and Taneja (1975) [92] presented an axiomatic characterization of degree $(r,s)$. It is as follows:

Let $H^{r,s}_n : \Delta_n$$\rightarrow I\!\!R$ be a real valued function satisfying

$$H^{r,s}_n(P)=\sum_{i=1}^n{h(p_i)},$$ and

$$\sum_{i=1}^n{\sum_{j=1}^m{h(p_iq_j)}}=\sum_{i=1}^n{\sum_{j=1}^m\,{p^r_i\,h(q_j)}}+\sum_{i=1}^n{\sum_{j=1}^m\,{q^s_j\,h(p_i)}},$$ (3.17)

where $h:[0,1] \rightarrow I\!\!R$ be a continuous function with$h({1\over 2})={1\over 2}$. Then

$$H^{r,s}_n(P)=(2^{1-r}-2^{1-s})^{-1}\sum_{i=1}^n{(p^r_i-p^s_i)},\r\neq s,\ r>0,\ s>0$$ (3.18)

Sharma and Taneja (1977) [93] extended the functional equation (3.17) by the following generalized additivity: 

$$H(P*Q)=G(P)H(Q)+H(P)G(Q),$$ for all $P \in \ \Delta_n$,$Q\ \in\ \Delta_m$ and $P*Q\in\Delta_{nm}$, where $$H(P)=\sum_{i=1}^n{h(p_i)},$$ and  $$G(P)=\sum_{i=1}^n{g(p_i)},$$ with $f$ and $g$ real valued continuous functions defined over [0,1] and $h({1\over 2})={1\over 2}$. This lead us to the measure (3.18) along with the trigonometric measures given in section 3.6.2.