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Properties of Unified (r,s)-Relative Information

The unified $(r,s)-$relative information${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q)$ given by (4.1) satisfies the following properties;

Property 4.1. (Continuity).${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q)$ is a continuous function of the pair $(P,Q)$ and is also continuous with respect to the parameters $r$ and $s$.

Property 4.2. (Symmetry).${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q)$ is a symmetric function of their arguments in the pair $(P,Q)$, i.e., 

$${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(p_1,...,p_n\vert\vert......}}^s_r(p_{\tau (1)},...,p_{\tau (n)}\vert\vert q_{\tau(1)},...,q_{\tau (n)}),$$

where $\tau$ is an arbitrary permutation of $1$ to $n$.

Property 4.3. (Expansibility). We can write 

$${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(p_1,...,p_n,0\vert\ve......{\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(p_1,...,p_n\vert\vert q_1,...,q_n).$$

Property 4.4.(Nonadditivity). We have

$${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P_1*P_2\vert\vert Q_1......vert\vert Q_1)\,{\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P_2\vert\vert Q_2),$$

for all $P_1,\ Q_{1}\in \Delta_n$, $P_2,\Q_{2} \in \Delta_m$ and $P_1*P_2$, $Q_1*Q_{2} \in \Delta_{nm}$.

Property 4.5. (Nonnegativity).${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q)\geq 0$ with equality iff $P=Q$.

Property 4.6. (Monotonicity).${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q)$ is an increasing function of $r$ ($s$ fixed) and of$s$ ($r$ fixed). In particular, when $r=s$, the result still holds.

Property 4.7. (Inequalities among the measures). We have

(i) ${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q) \left\{ \begin{arra......{\boldsymbol{\mathscr{D}}}}^1_r(P\vert\vert Q), & r\geq 1\end{array}\right.$

(ii) ${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q) \left\{ \begin{arra......\boldsymbol{\mathscr{D}}}}^s_1(P\vert\vert Q), &s\geq 1\end{array}\right.$

Property 4.8. (Convexity).$D^s_r(P\vert\vert Q)$ is a convex function of the pair of probability distributions $(P,Q) \in\Delta_n\times\Delta_n$ for $s\geq r> 0$.

Property 4.9. (Generalized data processing inequality). We have 

$${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r\big( P(B)\vert\vert Q(B)\big)\leq{\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q),$$

where $P(B)$ and $Q(B)$ are the probability distributions given by

$$P(B)=\Big(\sum_{i=1}^n{p_ib_{i1}},...,\sum_{i=1}^n{p_ib_{im}}\Big)\\in\ \Delta_m,$$ and $$Q(B)=\Big(\sum_{i=1}^n{q_ib_{i1}},...,\sum_{i=1}^n{q_ib_{im}}\Big)\\in\ \Delta_m,$$ where $B=\{b_{ij}\}$, $b_{ij}\geq 0$, $\forall \ i=1,2,...,n$; $j=1,2,...,m$ is a stochastic matrix such that$\sum_{j=1}^n{b_{ij}}=1$, $\forall \ i=1,2,...,n$.

Property 4.10. (Schur-convexity)${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q)$ is a Schur-convex function in the pair $(P,Q) \in\Delta_n\times\Delta_n$.

Property 4.11. For $1\leq\sigma \leq n$, we have 

$${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r\Big({1\over\sigma}\......},...,q_n\Big)\leq{\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q).$$

Property 4.12. (Order preserving) We have

(i) If $D(P_1\vert\vert Q_1)\geq D(P_2\vert\vert Q_2)$, then

$${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P_1\vert\vert Q_1)\geq {\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P_2\vert\vert Q_2),$$

for all $r\ \in\ (r_1,r_2)$, where $r_1$ and $r_2$ are determined by the equations: 

$$D^1_{r_1}(P_1\vert\vert Q_1)=D(P_2\vert\vert Q_2),$$

and $$D^1_{r_2}(P_2\vert\vert Q_2)=D(P_1\vert\vert Q_1).$$

(ii) If $D(P_1\vert\vert Q_1)\geq \log \big({p\over q}\big)_{2max}$, then

$${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P_1\vert\vert Q_1)\ge......nsuremath{\boldsymbol{\mathscr{D}}}}^s_r(P_2\vert\vert Q_2),\ 1\leq r <\infty$$

where  $$\big({p\over q}\big)_{2max} = \,\mathrel{\mathop{max}\limits_{i}}\big\{{p_{2i}\over q_{2i}}\big\}.$$