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Second Generalizations

The second generalizations of unified$(r,s)-I-$divergence measure based on an expression appearing in (4.8) is given by

$$^2{\ensuremath{\boldsymbol{\mathscr{I}}}}^s_r(P\vert\vert Q)=\lef......ert Q), & r\neq 1,\ \ s=1 \\ I(P\vert\vert Q), & r=1,\ \ s=1\end{array}\right.$$ (4.10)

where
$$^2I^s_r(P\vert\vert Q)=(1-2^{1-s})^{-1}\Bigg\{\sum_{i=1}^n{\Big...... 2}\Big) \Big( {p_i+q_i\over2}\Big)^{1-r} }-1\Bigg\},\ \ r\neq 1,\ \ s\neq 1,$$ $$^2I^s_1(P\vert\vert Q)=(1-2^{1-s})^{-1}\Big\{\exp_2 \big[ (s-1)I(P\vert\vert Q)\big] -1\Big\},\ \ s\neq 1$$ and $$^2R^s_r(P\vert\vert Q)=(r-1)^{-1}\log\Bigg\{ \sum_{i=1}^n{\Big({p^r_i+q^r_i\over 2}\Big) \Big( {p_i+q_i\over 2}\Big)^{1-r}}\Bigg\},\ \ r\neq 1.$$

The second generalizations of unified$(r,s)-J-$divergence measure is given by

$$^2{\ensuremath{\boldsymbol{\mathscr{J}}}}^s_r(P\vert\vert Q)=\lef......ert Q), & r\neq 1,\ \ s=1 \\ J(P\vert\vert Q), & r=1,\ \ s=1\end{array}\right.$$ (4.11)

where

$$^2J^s_r(P\vert\vert Q)=2(1-2^{1-s})^{-1}\Bigg\{\Big[\sum_{i=1}^......-r}_iq^r_i\over2}\Big)\Big]^{s-1\over r-1} }-1\Bigg\},\ \ r\neq 1,\ \ s\neq 1$$ $$^2J^s_1(P\vert\vert Q)=2(1-2^{1-s})^{-1}\Big\{\exp_2 \big[ \big({s-1\over2}\big) J(P\vert\vert Q)\big] -1\Big\},\ \ s\neq 1$$ and $$^2J^1_r(P\vert\vert Q)=2(r-1)^{-1}\log\Bigg\{ \sum_{i=1}^n{\Big({p^r_iq^{1-r}_i+p^{1-r}_iq^r_i\over 2}\Big) }\Bigg\},\ \ r\neq1.$$ In particular when r=s, we have  $$^1{\ensuremath{\boldsymbol{\mathscr{I}}}}^s_s(P\vert\vert Q)=\ ^2{\ensuremath{\boldsymbol{\mathscr{I}}}}^s_s(P\vert\vert Q),$$ and  $$^1{\ensuremath{\boldsymbol{\mathscr{J}}}}^s_s(P\vert\vert Q)=\ ^2{\ensuremath{\boldsymbol{\mathscr{J}}}}^s_s(P\vert\vert Q).$$

Note 4.6. Measures appearing in the unified expressions (4.6) and (4.7) are due to Taneja (1989) [105]. Most of the measures appearing in the unified expressions (4.10) and (4.11) are due to Taneja (1984b;1987;1989) [102], [103], [105] except the measures$^1J^1_r(P\vert\vert Q)$ and $J^s_s(P\vert\vert Q)$. The measure $^1J^1_r(P\vert\vert Q)$ is due to Burbea (1984) [17] and the measure $J^s_s(P\vert\vert Q)$ is due to Rathie and Sheng (1981) [80] and Burbea and Rao (1982a;b) [18], [19].