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Bivariate Cases

In this subsection, we shall give properties of the generalized conditional entropies for the bivariate case.

Property 6.1. For $r,s \in (-\infty,\infty)$, we have 

$${\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X,Y)\geq {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X)\$$   or$$\ {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(Y).$$

Property 6.2. We have  $$^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)\geq 0 \,\,(\alpha=1,2,3\,\,$$   and$$\,\, 4).$$

Property 6.3. If $X$ and $Y$ are independent random variables, then 

$${\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X,Y)={\ensuremath{\bo......ldsymbol{\mathscr{H}}}}^s_r(X)\,{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(Y).$$

Property 6.4. For any $X$ and $Y$, we have

${\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(Y)+\ ^2{\ensuremath{\boldsymbol{\m......mathscr{H}}}}^s_r(Y)\,\,^2{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)$

$$=(2^{1-s}-1)^{-1}\{\exp_2[(1-s)({\ensuremath{\boldsymbol{\mathscr{H}}}}^1_r(Y)+{\ensuremath{\boldsymbol{\mathscr{H}}}}^1_r(X\vert Y))]-1\}.$$

Property 6.5. The following inequalities hold:

(i)$^2{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)\left\{\begin{array}{ll......ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y),& r<0\end{array}\right.$

(ii)$^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)\left\{\begin{array}{ll......remath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y),& s\leq r\end{array}\right.$

(iii)${\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X,Y)\left\{\begin{array}{ll} \leq ......ert Y),& r{{s-1}\over {r-1}}\leq 1\leq {{s-1}\over {r-1}}\end{array}\right.$

(iv)$^4{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)\left\{\begin{array}{ll......rt Y),& r{{s-1}\over {r-1}}\leq 1\leq{{s-1}\over {r-1}}\end{array}\right.$

Note 6.2. The conditions $\frac{s-1}{r-1} \leq 1 \leq r\frac{s-1}{r-1}$ are equivalent to either $r \geq s \geq2-\frac{1}{r} \geq 1$ or $r<0$, $s \geq 2-\frac{1}{r}$, and the conditions $\frac{s-1}{r-1} \geq 1 \geq r \frac{s-1}{r-1}$ are equivalent to either $1 \geq r \geq s \geq 2-\frac{1}{r}$, $r>0$ or $s \leq r < 0$.

Property 6.6. We have

(i) $^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)\left\{\begin{array}{ll......\ \mbox{with}\s\leq 1\ \mbox{or}\ 1<s\leq 2-{1\over r};\end{array}\right.$

(ii) $^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)\left\{\begin{arra......eq\ {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X),& r<0,\end{array}\right.$ for $\alpha$=2 and 3;

(iii) $^4{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)\left\{\begin{array}{ll......suremath{\boldsymbol{\mathscr{H}}}}^s_r(X),& s\leq r < 0.\end{array}\right.$

The equality sign holds iff $X$ and $Y$ are independent random variables.

As a consequence of properties 6.5 and 6.6, we have the following property:

Property 6.7. (i) For $s \leq r < 0$, we have

$$0\leq {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X)\leq\ ^3{\ens......}}}^s_r(X\vert Y)\leq\ ^4{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y).$$

(ii) For $s\geq r> 0$, we have 

$$0 \leq \, ^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)......athscr{H}}}}^s_r(X\vert Y)\leq{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X).$$

(iii) For $r \geq s \geq2-\frac{1}{r} \geq 1$, we have 

$$0\leq \ ^4{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)\l......athscr{H}}}}^s_r(X\vert Y)\leq{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X).$$

(iv) For $r<0,\,\, s\geq 2-\frac{1}{r}$, we have 

$$0\leq \ ^4{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)\l......}}^s_r(X\vert Y)\leq\^2{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y).$$