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First Generalizations

Replacing $D(P\vert\vert Q)$ by ${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q)$ in the expressions (2.6) and (2.4),we get

$$^1{\ensuremath{\boldsymbol{\mathscr{I}}}}^s_r(P\vert\vert Q)= {1\......uremath{\boldsymbol{\mathscr{D}}}}^s_r\big( Q\vert\vert{P+Q\over2}\big)\Bigg],$$ (4.6)

and

$$^1{\ensuremath{\boldsymbol{\mathscr{J}}}}^s_r(P\vert\vert Q)= {\e......^s_r(P\vert\vert Q)+{\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(Q\vert\vert P),$$ (4.7)

respectively.

The unified $(r,s)-$divergence measures according to the expression (4.6) is given by

$$^1{\ensuremath{\boldsymbol{\mathscr{I}}}}^s_r(P\vert\vert Q)=\lef......), & r\neq 1,\ \ s=1 \\ I(P\vert\vert Q), & r=1,\ \s=1\end{array}\right.$$ where

$$^1I^s_r(P\vert\vert Q)=(1-2^{1-s})^{-1}\Bigg\{ \Big[ \sum_{i=1}^n{p^r_i\Big( {p_i+q_i\over 2}\Big) ^{1-r}}\Big]^{s-1\over r-1}+$$

$$\Big[\sum_{i=1}^n{q^r_i \Big( {p_i+q_i\over 2}\Big)^{1-r}}\Big]^{s-1\over r-1} -2\Bigg\},\ r\neq 1,\ s\neq 1$$ $$^1I^s_1(P\vert\vert Q)=(1-2^{1-s})^{-1}\Big\{ \exp_2\Big[ (s-1)\sum_{i=1}^n{p_i\,\log \Big( {2p_i\over p_i+q_i}\Big)}\Big]+$$ $$\exp_2\Big[ (s-1)\sum_{i=1}^n{q_i\,\log \Big( {2q_i\overp_i+q_i}\Big)} \Big]-2 \Big\},\ s\neq 1$$ and $$^1I^1_r(P\vert\vert Q)=\big[ 2(r-1)\big]^{-1}\log\Big\{ \Big[\s......\sum_{i=1}^n{q^r_i \Big( {p_i+q_i\over 2}\Big) ^{1-r}}\Big]\Big\},\ r\neq 1$$

The unified $(r,s)-$divergence measures according to the expression (4.7) is given by

$$^1{\ensuremath{\boldsymbol{\mathscr{J}}}}^s_r(P\vert\vert Q)=\lef......), & r\neq 1,\ \ s=1 \\ J(P\vert\vert Q), & r=1,\ \s=1\end{array}\right.$$ where

$$^1J^s_r(P\vert\vert Q)=(1-2^{1-s})^{-1}\Bigg\{\Big( \sum_{i=1}^n{p^r_iq^{1-r}_i}\Big)^{s-1\over r-1}+$$

$$\Big( \sum_{i=1}^n{p^r_iq^{1-r}_i}\Big)^{s-1\over r-1} -2\Bigg\},\ r\neq 1,\ s\neq 1$$

$$^1J^s_1(P\vert\vert Q)=(1-2^{1-s})^{-1}\Big\{\exp_2 \Big( (s-1)\sum_{i=1}^n{p_i \log{p_i\over q_i}}\Big)+$$

$$\exp_2\Big((s-1)\sum_{i=1}^n{q_i\log{q_i\over p_i}}\Big) -2\Big\},\ s\neq 1$$ and

$$^1J^1_r(P\vert\vert Q)=(r-1)^{-1}\log\Big\{\Big( \sum_{i=1}^n{p^r_i q^{1-r}_i}\Big) \Big(\sum_{i=1}^n{q^r_ip^{1-r}_i}\Big)\Big\},\ r\neq 1$$

In particular, when $r=s$, we have

$^1I^s_s(P\vert\vert Q) = I^s_s(P\vert\vert Q)$

$$= (1-2^{1-s})^{-1} \Big\{\sum_{i=1}^n{\Big({p^s_i+q^s_i\over2}\Big)\Big({p_i+q_i\over 2}\Big)^{1-s}-1}\Big\},\ s\neq 1,\ s>0$$ (4.8)

and

$^1J^s_s(P\vert\vert Q)=J^s_s(P\vert\vert Q)$

$$= (1-2^{1-s})^{-1} \Big\{\sum_{i=1}^n{\big(p^s_iq^{1-s}_i+p^{1-s}_iq^s_i\big)}-2\Big\},\ s\neq 1,\ s>0$$ (4.9)

The expressions appearing in (4.8) and (4.9) shall be used to give an alternative way for generalizing unified $(r,s)-$divergence measures.