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M-Dimensional Unified (r,s)-Arithmetic and Geometric Mean Divergences

The three different ways of defining unified$(r,s)-A$ & $G-$divergences or unified $(r,s)-T-$divergence measures are given by

$$^\alpha{\ensuremath{\boldsymbol{\mathscr{T}}}}^s_r(P_1,P_2,...,P_......P_M), & r\neq 1,\ \ s=1 \\ T(P_1,P_2,...,P_M), &r=1,\ \ s=1\end{array}\right.$$ (5.23)

$\alpha=1,2$ and 3, where for $\alpha =1$, we have

$^1T^s_r(P_1,P_2,...,P_M)$

$$=(1-2^{1-s})^{-1}\Big\{\Big[ \sum_{i=1}^n{ \Big(\sum_{j=1}^M{......lambda_kp^{1-r}_{ki}}\Big)\Big]^{{s-1}\over{r-1}}}-1\Big\},\r\neq 1,\ s\neq 1$$ $$^1T^s_1(P_1,P_2,...,P_M)=(1-2^{1-s})^{-1}\Big\{2^{(s-1)T(P_1,P_2,...,P_M)}-1\Big\},\, s\neq 1$$ and $$^1T^1_r(P_1,P_2,...,P_M)=(r-1)^{-1}\log_2\Big\{\sum_{i=1}^n{\Big(......_{ji}}\Big)^r\Big(\sum_{k=1}^M{\lambda_kp^{1-r}_{ki}}\Big)}\Big\}, \, r\neq 1$$ For $\alpha=2$, we have

$$^2{\ensuremath{\boldsymbol{\mathscr{T}}}}^s_r(P_1,P_2,...,P_M)=\s......dsymbol{\mathscr{D}}}}^s_r \Big(\sum_{j=1}^M{\lambda_jP_j}\vert\vert P_k \Big).$$ (5.24)

For $\alpha =3$, we have

$^3 T^s_r(P_1,P_2,...,P_M)$

$$=(1-2^{1-s})^{-1}\Big\{\Big[\sum_{i=1}^n{\Big(\sum_{j=1}^M{\l......lambda_k}}\Big)^{1-r}\Big]^{{s-1}\over{r-1}}}-1\Big\}, \, r\neq 1, \, s\neq 1$$ $$^3T^s_1(P_1,P_2,...,P_M)=(1-2^{1-s})^{-1}\Big\{2^{(s-1)T(P_1,P_2,...,P_M)}-1\Big\},\, s\neq 1$$ and $$^3T^1_r(P_1,P_2,...,P_M)=(r-1)^{-1}\log_2\Big\{\sum_{i=1}^n{\Big(......}\Big)^r\Big(\prod_{k=1}^M{p_{ki}^{\lambda_k}}\Big)^{1-r}} \Big\}, \, r\neq 1$$ In particular, we have 

$$^1 {\ensuremath{\boldsymbol{\mathscr{T}}}}^s_r(P_1,P_2,...,P_M)=\......(P_1,P_2,...,P_M)= {\ensuremath{\boldsymbol{\mathscr{T}}}}^s_s(P_1,P_2,...,P_M)$$

where $${\ensuremath{\boldsymbol{\mathscr{T}}}}^s_s(P_1,P_2,...,P_M)=\lef......1,P_2,...,P_M), & \ s\neq 1 \\ T(P_1,P_2,...,P_M), & s=1\end{array}\right.$$ with  $$T^s_s(P_1,P_2,...,P_M)=(1-2^{1-s})^{-1}\Big\{\sum_{i=1}^n{ \Big......ij}}\Big)^s\Big(\sum_{k=1}^M{\lambda_k p^{1-s}_{ki}}\Big)}-1\Big\},\ s\neq1$$ For two dimensional case, i.e., when M=2,$\lambda_1=\lambda_2={1\over 2},\ P_1 =P$ and $P_2=Q$, we have

$^1 T^s_r(P_1,P_2)=\ ^1T^s_r(P\vert\vert Q)$

$$= (1-2^{1-s})^{-1}\Big\{ \Big[ \sum_{i=1}^n {\Big( {{p_i+q_i}\ove......g)^r\Big( {{p^{1-r}_i+q^{1-r}_i}\over 2}\Big)}\Big]^{{s-1}\over{r-1}}-1\Big\},$$ (5.25)

$^2 T^s_r(P_1,P_2)=\ ^2T^s_r(P\vert\vert Q)$

$$= (1-2^{1-s})^{-1}{1\over 2}\Big\{ \Big[ \sum_{i=1}^n {\Big({{p_......^n {\Big( {{p_i+q_i}\over 2}\Big)^r p^{1-r}_i}\Big]^{{s-1}\over{r-1}}-2 \Big\},$$ (5.26)

and

$^3 T^s_r(P_1,P_2)=\ ^3T^s_r(P\vert\vert Q)$

$$= (1-2^{1-s})^{-1}\Big\{ \Big[ \sum_{i=1}^n{ \Big( {{p_i+q_i}\over2} \Big)^r\Big(\sqrt{p_iq_i}\Big)^{1-r}}\Big]^{{s-1}\over{r-1}}-1\Big\},$$ (5.27)