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Composition Relations Among the Unified (r,s)-Entropy Measures

We can write  $${\bf\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P)={\bf\ensuremath......ol{\mathscr{M}}}}_s\big({\bf\ensuremath{\boldsymbol{\mathscr{H}}}}^1_r(P)\big),$$ where

$${\bf {\ensuremath{\boldsymbol{\mathscr{M}}}}}_s(x)=\left\{\begin{......1-s}-1)}^{-1}\big[(2^{(1-s)x}-1)\big], & s\neq 1 \\ x, & s=1\end{array}\right.$$ (3.10)

for all $x\geq 0$.

We can write 

$${\bf {\ensuremath{\boldsymbol{\mathscr{M}}}}}_s(x)=\theta_s(2^{-x})$$ where $\theta_s(x)$ is as given by

$$\theta_s(x)=\left\{\begin{array}{ll}{(2^{1-s}-1)}^{-1}\big[x^{s-1}-1\big], & s\neq 1 \\ -\log_2x, &s=1\end{array}\right.$$ (3.11)

for all $s \in (-\infty,\infty)$ and $x\ \varepsilon\ (0,\infty)$, for $t<0$ in (3.11), we have $v_i>0,\ \forall i$.

The functions ${\ensuremath{\boldsymbol{\mathscr{M}}}}(x)$ and $\theta_s(x)$ satisfy some interesting properties given in Taneja (1995) [108].