**4. Pseudo-Additivity of the Weighted Tsallis and Kaniadakis Divergences**

Let (Ω, T , *μi*), *i* = 1, 2 be two measure spaces and *λ*<sup>1</sup> a measure on (Ω, T ) such that *μ<sup>i</sup>* ∼ *λ*<sup>1</sup> for any *i* = 1, 2. We consider Radon–Nikodým derivatives *f* (1) <sup>1</sup> and *f* (1) <sup>2</sup> on Ω, i.e., *f* (1) *<sup>i</sup>* <sup>=</sup> *<sup>d</sup>μ<sup>i</sup> dλ*<sup>1</sup> for any *i* = 1, 2. Let also (*S*, S, *νj*), *j* = 1, 2 be two measure spaces and *λ*<sup>2</sup> a measure on (*S*, S) such that *ν<sup>j</sup>* ∼ *λ*<sup>2</sup> for any *j* = 1, 2. We apply Radon–Nikodým Theorem and find the non-negative measurable functions *f* (2) <sup>1</sup> and *f* (2) <sup>2</sup> defined on *S* such that *f* (2) *<sup>j</sup>* <sup>=</sup> *<sup>d</sup>ν<sup>j</sup> dλ*<sup>2</sup> for any *j* = 1, 2. We take *w*<sup>1</sup> : Ω → (0, ∞) and *w*<sup>2</sup> : *S* → (0, ∞) two weight functions.

We consider the measure *λ* on (Ω × *S*, T ×S) induced by *λ*<sup>1</sup> and *λ*2. Because *μ<sup>i</sup>* × *ν<sup>i</sup>* is absolutely continuous with respect to *λ*, we apply Radon–Nikodým Theorem and find two non-negative measurable functions *<sup>f</sup>*<sup>1</sup> and *<sup>f</sup>*<sup>2</sup> on <sup>Ω</sup> <sup>×</sup> *<sup>S</sup>* such that *fi* <sup>=</sup> *<sup>d</sup>*(*μ<sup>i</sup>* <sup>×</sup> *<sup>ν</sup>i*) *<sup>d</sup><sup>λ</sup>* for any *i* = 1, 2.

The uniqueness from Radon–Nikodým Theorem assures us that *fi*(*ω*,*s*) = *f* (1) *<sup>i</sup>* (*ω*)*f* (2) *<sup>i</sup>* (*s*) for any *ω* ∈ Ω, *s* ∈ *S* and any *i* = 1, 2.

Let *A* ∈ T and *B* ∈ S.

We define the weighted Tsallis divergence for product measures via

$$\frac{D\_{k}^{w\_{1}w\_{2},T}(\mu\_{1}\times\nu\_{1}|\mu\_{2}\times\nu\_{2},A\times B)}{\iint\_{A\times B} w\_{1}w\_{2}d(\mu\_{1}\times\nu\_{1})} \iint\_{A\times B} w\_{1}(\omega)w\_{2}(s)\log\_{k}^{T}\left(\frac{f\_{1}(\omega,s)}{f\_{2}(\omega,s)}\right)d(\mu\_{1}\times\nu\_{1})(\omega,s)=0}$$

$$\frac{1}{\int\_{A} w\_{1}d\mu\_{1}\int\_{B} w\_{2}d\nu\_{1}} \iint\_{A\times B} w\_{1}(\omega)w\_{2}(s)f\_{1}(\omega,s)\log\_{k}^{T}\left(\frac{f\_{1}(\omega,s)}{f\_{2}(\omega,s)}\right)d\lambda(\omega,s) \text{ if }$$

$$(\mu\_{1}\times\nu\_{1})(A\times B)\neq 0$$

and

$$D\_k^{w\_1 w\_2, T}(\mu\_1 \times \nu\_1 | \mu\_2 \times \nu\_2, A \times B) = 0 \text{ if } (\mu\_1 \times \nu\_1)(A \times B) = 0.$$

The weighted Kaniadakis divergence for product measures is given by

$$\frac{D\_{k}^{w\_{1}w\_{2},K}(\mu\_{1}\times\nu\_{1}|\mu\_{2}\times\nu\_{2},A\times B)}{\iint\_{A\times B} w\_{1}w\_{2}d(\mu\_{1}\times\nu\_{1})} \iint\_{A\times B} w\_{1}(\omega)w\_{2}(s)\log\_{k}^{K}\left(\frac{f\_{1}(\omega,s)}{f\_{2}(\omega,s)}\right)d(\mu\_{1}\times\nu\_{1})(\omega,s)=0}$$

$$\frac{1}{\int\_{A} w\_{1}d\mu\_{1}\int\_{B} w\_{2}d\nu\_{1}} \iint\_{A\times B} w\_{1}(\omega)w\_{2}(s)f\_{1}(\omega,s)\log\_{k}^{K}\left(\frac{f\_{1}(\omega,s)}{f\_{2}(\omega,s)}\right)d\lambda(\omega,s) \text{ if }$$

$$(\mu\_{1}\times\nu\_{1})(A\times B)\neq 0$$

and

$$D\_k^{w\_1 w\_2, K}(\mu\_1 \times \nu\_1 | \mu\_2 \times \nu\_2, A \times B) = 0 \text{ if } (\mu\_1 \times \nu\_1)(A \times B) = 0.$$

**Lemma 2** (see [41])**.** *We have the following pseudo-additivity property for Tsallis logarithm (valid for any x*, *y* > 0*):*

$$
\log\_k^T(\mathbf{x}y) = \log\_k^T \mathbf{x} + \log\_k^T y + k(\log\_k^T \mathbf{x})(\log\_k^T y).
$$

**Theorem 3.** *Let A* ∈ T *and B* ∈ S *such that* (*μ*<sup>1</sup> × *ν*1)(*A* × *B*) = 0*. The weighted Tsallis and Kaniadakis divergences for product measures satisfy the following pseudo-additivity properties:*

(*a*) *Dw*1*w*2,*<sup>T</sup> <sup>k</sup>* (*μ*<sup>1</sup> <sup>×</sup> *<sup>ν</sup>*1|*μ*<sup>2</sup> <sup>×</sup> *<sup>ν</sup>*2, *<sup>A</sup>* <sup>×</sup> *<sup>B</sup>*) = *<sup>D</sup>w*1,*<sup>T</sup> <sup>k</sup>* (*μ*1|*μ*2, *<sup>A</sup>*) + *<sup>D</sup>w*2,*<sup>T</sup> <sup>k</sup>* (*ν*1|*ν*2, *B*) + *kDw*1,*<sup>T</sup> <sup>k</sup>* (*μ*1|*μ*2, *<sup>A</sup>*)*Dw*2,*<sup>T</sup> <sup>k</sup>* (*ν*1|*ν*2, *B*)*.* (*b*) *Dw*1*w*2,*<sup>K</sup> <sup>k</sup>* (*μ*<sup>1</sup> <sup>×</sup> *<sup>ν</sup>*1|*μ*<sup>2</sup> <sup>×</sup> *<sup>ν</sup>*2, *<sup>A</sup>* <sup>×</sup> *<sup>B</sup>*) = *<sup>D</sup>w*1,*<sup>K</sup> <sup>k</sup>* (*μ*1|*μ*2, *<sup>A</sup>*) + *<sup>D</sup>w*2,*<sup>K</sup> <sup>k</sup>* (*ν*1|*ν*2, *B*) + *k* 2 - *Dw*1,*<sup>T</sup> <sup>k</sup>* (*μ*1|*μ*2, *<sup>A</sup>*)*Dw*2,*<sup>T</sup> <sup>k</sup>* (*ν*1|*ν*2, *<sup>B</sup>*) <sup>−</sup> *<sup>D</sup>w*1,*<sup>T</sup>* <sup>−</sup>*<sup>k</sup>* (*μ*1|*μ*2, *<sup>A</sup>*)*Dw*2,*<sup>T</sup>* <sup>−</sup>*<sup>k</sup>* (*ν*1|*ν*2, *<sup>B</sup>*) *.* (*c*) *Dw*1*w*2,*<sup>K</sup> <sup>k</sup>* (*μ*<sup>1</sup> <sup>×</sup> *<sup>ν</sup>*1|*μ*<sup>2</sup> <sup>×</sup> *<sup>ν</sup>*2, *<sup>A</sup>* <sup>×</sup> *<sup>B</sup>*) = <sup>1</sup> *B w*2*dν*<sup>1</sup> ⎛ ⎝ *B w*2(*s*)*f* (2) <sup>1</sup> (*s*) *f* (2) <sup>1</sup> (*s*) *f* (2) <sup>2</sup> (*s*) *k dλ*2(*s*) ⎞ ⎠ · *Dw*1,*<sup>K</sup> <sup>k</sup>* (*μ*1|*μ*2, *<sup>A</sup>*) + <sup>1</sup> *A w*1*dμ*<sup>1</sup> ⎛ ⎝ *A w*1(*ω*)*f* (1) <sup>1</sup> (*ω*) *f* (1) <sup>1</sup> (*ω*) *f* (1) <sup>2</sup> (*ω*) −*<sup>k</sup> dλ*1(*ω*) ⎞ ⎠*Dw*2,*<sup>K</sup> <sup>k</sup>* (*ν*1|*ν*2, *B*)*.*

**Proof.** (*a*) According to Lemma 2, we have

*Dw*1*w*2,*<sup>T</sup> <sup>k</sup>* (*μ*<sup>1</sup> × *ν*1|*μ*<sup>2</sup> × *ν*2, *A* × *B*) = 1 *A w*1*dμ*<sup>1</sup> *B w*2*dν*<sup>1</sup> *A*×*B w*1(*ω*)*w*2(*s*)*f*1(*ω*,*s*)log*<sup>T</sup> k f*1(*ω*,*s*) *f*2(*ω*,*s*) *dλ*(*ω*,*s*) = 1 *A w*1*dμ*<sup>1</sup> · <sup>1</sup> *B w*2*dν*<sup>1</sup> *A*×*B w*1(*ω*)*w*2(*s*)*f* (1) <sup>1</sup> (*ω*)*f* (2) <sup>1</sup> (*s*)log*<sup>T</sup> k f* (1) <sup>1</sup> (*ω*) *f* (1) <sup>2</sup> (*ω*) *dλ*1(*ω*)*dλ*2(*s*) + 1 *A w*1*dμ*<sup>1</sup> · <sup>1</sup> *B w*2*dν*<sup>1</sup> *A*×*B w*1(*ω*)*w*2(*s*)*f* (1) <sup>1</sup> (*ω*)*f* (2) <sup>1</sup> (*s*)log*<sup>T</sup> k f* (2) <sup>1</sup> (*s*) *f* (2) <sup>2</sup> (*s*) *dλ*1(*ω*)*dλ*2(*s*) + *k* · 1 *A w*1*dμ*<sup>1</sup> · <sup>1</sup> *B w*2*dν*<sup>1</sup> · *A*×*B w*1(*ω*)*w*2(*s*)*f* (1) <sup>1</sup> (*ω*)*f* (2) <sup>1</sup> (*s*) log*<sup>T</sup> k f* (1) <sup>1</sup> (*ω*) *f* (1) <sup>2</sup> (*ω*) log*<sup>T</sup> k f* (2) <sup>1</sup> (*s*) *f* (2) <sup>2</sup> (*s*) *dλ*1(*ω*)*dλ*2(*s*) = 1 *A w*1*dμ*<sup>1</sup> *A w*1(*ω*)*f* (1) <sup>1</sup> (*ω*)log*<sup>T</sup> k f* (1) <sup>1</sup> (*ω*) *f* (1) <sup>2</sup> (*ω*) *dλ*1(*ω*) 1 *B w*2*dν*<sup>1</sup> · *B w*2(*s*)*f* (2) <sup>1</sup> (*s*)*dλ*2(*s*) +

1 *A w*1*dμ*<sup>1</sup> *A w*1(*ω*)*f* (1) <sup>1</sup> (*ω*)*dλ*1(*ω*) 1 *B w*2*dν*<sup>1</sup> *B w*2(*s*)*f* (2) <sup>1</sup> (*s*)log*<sup>T</sup> k f* (2) <sup>1</sup> (*s*) *f* (2) <sup>2</sup> (*s*) *dλ*2(*s*) + *<sup>k</sup>* · <sup>1</sup> *A w*1*dμ*<sup>1</sup> *A w*1(*ω*)*f* (1) <sup>1</sup> (*ω*)log*<sup>T</sup> k f* (1) <sup>1</sup> (*ω*) *f* (1) <sup>2</sup> (*ω*) *dλ*1(*ω*) 1 *B w*2*dν*<sup>1</sup> · *B w*2(*s*)*f* (2) <sup>1</sup> (*s*)log*<sup>T</sup> k f* (2) <sup>1</sup> (*s*) *f* (2) <sup>2</sup> (*s*) *dλ*2(*s*) = *Dw*1,*<sup>T</sup> <sup>k</sup>* (*μ*1|*μ*2, *<sup>A</sup>*) + *<sup>D</sup>w*2,*<sup>T</sup> <sup>k</sup>* (*ν*1|*ν*2, *<sup>B</sup>*) + *kDw*1,*<sup>T</sup> <sup>k</sup>* (*μ*1|*μ*2, *<sup>A</sup>*)*Dw*2,*<sup>T</sup> <sup>k</sup>* (*ν*1|*ν*2, *B*). (*b*) Because log*<sup>K</sup> <sup>k</sup> <sup>x</sup>* <sup>=</sup> <sup>1</sup> 2 - log*<sup>T</sup> <sup>k</sup> <sup>x</sup>* <sup>+</sup> log*<sup>T</sup>* <sup>−</sup>*<sup>k</sup> <sup>x</sup>* , we have *Dw*1*w*2,*<sup>K</sup> <sup>k</sup>* (*μ*<sup>1</sup> × *ν*1|*μ*<sup>2</sup> × *ν*2, *A* × *B*) = 1 *A w*1*dμ*<sup>1</sup> *B w*2*dν*<sup>1</sup> *A*×*B w*1(*ω*)*w*2(*s*)*f*1(*ω*,*s*)log*<sup>K</sup> k f*1(*ω*,*s*) *f*2(*ω*,*s*) *dλ*(*ω*,*s*) = 1 *A w*1*dμ*<sup>1</sup> · <sup>1</sup> *B w*2*dν*<sup>1</sup> · *A*×*B w*1(*ω*)*w*2(*s*)*f* (1) <sup>1</sup> (*ω*)*f* (2) <sup>1</sup> (*s*) · 1 2 log*<sup>T</sup> k f* (1) <sup>1</sup> (*ω*)*f* (2) <sup>1</sup> (*s*) *f* (1) <sup>2</sup> (*ω*)*f* (2) <sup>2</sup> (*s*) + log*<sup>T</sup>* −*k f* (1) <sup>1</sup> (*ω*)*f* (2) <sup>1</sup> (*s*) *f* (1) <sup>2</sup> (*ω*)*f* (2) <sup>2</sup> (*s*) *dλ*1(*ω*)*dλ*2(*s*) = 1 2 - *Dw*1*w*2,*<sup>T</sup> <sup>k</sup>* (*μ*<sup>1</sup> <sup>×</sup> *<sup>ν</sup>*1|*μ*<sup>2</sup> <sup>×</sup> *<sup>ν</sup>*2, *<sup>A</sup>* <sup>×</sup> *<sup>B</sup>*) + *<sup>D</sup>w*1*w*2,*<sup>T</sup>* <sup>−</sup>*<sup>k</sup>* (*μ*<sup>1</sup> <sup>×</sup> *<sup>ν</sup>*1|*μ*<sup>2</sup> <sup>×</sup> *<sup>ν</sup>*2, *<sup>A</sup>* <sup>×</sup> *<sup>B</sup>*) .

Using (*a*), we get

$$\begin{split} &D\_{k}^{w\_{1}w\_{2},K}(\mu\_{1}\times\nu\_{1}|\mu\_{2}\times\nu\_{2},A\times B) = \\ &\frac{1}{2}\Big(D\_{k}^{w\_{1},T}(\mu\_{1}|\mu\_{2},A) + D\_{k}^{w\_{2},T}(\nu\_{1}|\nu\_{2},B) + kD\_{k}^{w\_{1},T}(\mu\_{1}|\mu\_{2},A)D\_{k}^{w\_{2},T}(\nu\_{1}|\nu\_{2},B)\Big) + \\ &\frac{1}{2}\Big(D\_{-k}^{w\_{1},T}(\mu\_{1}|\mu\_{2},A) + D\_{-k}^{w\_{2},T}(\nu\_{1}|\nu\_{2},B) - kD\_{-k}^{w\_{1},T}(\mu\_{1}|\mu\_{2},A)D\_{\mathop{w\_{2}},T}^{w\_{2},T}(\nu\_{1}|\nu\_{2},B)\Big) = \\ &D\_{k}^{w\_{1},K}(\mu\_{1}|\mu\_{2},A) + D\_{k}^{w\_{2},K}(\nu\_{1}|\nu\_{2},B) + \\ &\frac{k}{2}\Big(D\_{k}^{w\_{1},T}(\mu\_{1}|\mu\_{2},A)D\_{k}^{w\_{2},T}(\nu\_{1}|\nu\_{2},B) - D\_{-k}^{w\_{1},T}(\mu\_{1}|\mu\_{2},A)D\_{-k}^{w\_{2},T}(\nu\_{1}|\nu\_{2},B)\Big). \end{split}$$

(*c*) It is easy to prove that

log*<sup>K</sup> <sup>k</sup>* (*xy*) = *<sup>y</sup><sup>k</sup>* · *<sup>x</sup><sup>k</sup>* <sup>−</sup> *<sup>x</sup>*−*<sup>k</sup>* <sup>2</sup>*<sup>k</sup>* <sup>+</sup> *<sup>x</sup>*−*<sup>k</sup>* · *<sup>y</sup><sup>k</sup>* <sup>−</sup> *<sup>y</sup>*−*<sup>k</sup>* <sup>2</sup>*<sup>k</sup>* <sup>=</sup> *<sup>y</sup><sup>k</sup>* log*<sup>K</sup> <sup>k</sup> <sup>x</sup>* <sup>+</sup> *<sup>x</sup>*−*<sup>k</sup>* log*<sup>K</sup> <sup>k</sup> y* for any *x*, *y* ∈ (0, ∞). Hence,

$$\begin{split} \frac{D\_{k}^{w\_{1}w\_{2},K}(\mu\_{1}\times v\_{1}|\mu\_{2}\times v\_{2},A\times B)}{\int\_{A}w\_{1}d\mu\_{1}\int\_{B}w\_{2}d\nu\_{1}} \iint\_{A\times B}w\_{1}(\omega)w\_{2}(s)f\_{1}(\omega,s)\log\_{k}^{K}\left(\frac{f\_{1}(\omega,s)}{f\_{2}(\omega,s)}\right)d\lambda(\omega,s) = \\ \frac{1}{\int\_{A}w\_{1}d\mu\_{1}} \cdot \frac{1}{\int\_{B}w\_{2}d\nu\_{1}} \iint\_{A\times B}w\_{1}(\omega)w\_{2}(s)f\_{1}^{(1)}(\omega)f\_{1}^{(2)}(\omega) \left[\left(\frac{f\_{1}^{(2)}(s)}{f\_{2}^{(2)}(s)}\right)^{k}\log\_{k}^{K}\left(\frac{f\_{1}^{(1)}(\omega)}{f\_{2}^{(1)}(\omega)}\right) + 1\right] \\ \frac{\left(f\_{1}^{(1)}(\omega)\right)^{-k}}{f\_{2}^{(1)}(\omega)} \log\_{k}^{K}\left(\frac{f\_{1}^{(2)}(s)}{f\_{2}^{(2)}(s)}\right)d\lambda\_{1}(\omega)d\lambda\_{2}(s) = \\ \frac{1}{\int\_{A}w\_{1}d\mu\_{1}} \left(\int\_{A}w\_{1}(\omega)f\_{1}^{(1)}(\omega)\log\_{k}^{K}\left(\frac{f\_{1}^{(1)}(\omega)}{f\_{2}^{(1)}(\omega)}\right)d\lambda\_{1}(\omega)\right). \end{split}$$

$$\begin{split} &\frac{1}{\int\_{B} w\_{2} d\nu\_{1}} \left( \int\_{B} w\_{2}(s) f\_{1}^{(2)}(s) \left( \frac{f\_{1}^{(2)}(s)}{f\_{2}^{(2)}(s)} \right)^{k} d\lambda\_{2}(s) \right) + \\ &\frac{1}{\int\_{A} w\_{1} d\mu\_{1}} \left( \int\_{A} w\_{1}(\omega) f\_{1}^{(1)}(\omega) \left( \frac{f\_{1}^{(1)}(\omega)}{f\_{2}^{(1)}(\omega)} \right)^{-k} d\lambda\_{1}(\omega) \right) \\ &\quad \frac{1}{\int\_{B} w\_{2} d\nu\_{1}} \left( \int\_{B} w\_{2}(s) f\_{1}^{(2)}(s) \log \frac{f\_{1}^{(2)}(s)}{f\_{2}^{(2)}(s)} \right) d\lambda\_{2}(s) \right) = \\ &\frac{1}{\int\_{B} w\_{2} d\nu\_{1}} \left( \int\_{B} w\_{2}(s) f\_{1}^{(2)}(s) \left( \frac{f\_{1}^{(2)}(s)}{f\_{2}^{(2)}(s)} \right)^{k} d\lambda\_{2}(s) \right) D\_{k}^{w\_{1},K}(\mu\_{1}|\mu\_{2},A) + \\ &\quad \frac{1}{\int\_{A} w\_{1} d\mu\_{1}} \left( \int\_{A} w\_{1}(\omega) f\_{1}^{(1)}(\omega) \left( \frac{f\_{1}^{(1)}(\omega)}{f\_{2}^{(1)}(\omega)} \right)^{-k} d\lambda\_{1}(\omega) \right) D\_{k}^{w\_{2},K}(\nu\_{1}|\nu\_{2},B). \end{split}$$

#### **5. Conclusions**

With the help of some inequalities concerning Tsallis logarithm, we obtained inequalities between the weighted Tsallis and Kaniadakis divergences on an arbitrary measurable non-negligible set and Tsallis logarithm, respectively, Kaniadakis logarithm (Theorem 1). We showed that the aforementioned divergences are limited by similar bounds with those that limit Kullback–Leibler divergence (Theorem 2) and proved that are pseudo-additive (Theorem 3).

**Author Contributions:** Conceptualization, R.-C.S., S.-C.S. and V.P.; Formal analysis, V.P.; Investigation, R.-C.S., S.-C.S. and V.P.; Methodology, R.-C.S.; Validation, S.-C.S.; Writing—original draft, R.-C.S.; Writing—review and editing, S.-C.S. and V.P. All authors contributed equally to the paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors are very much indebted to the anonymous referees and to the editors for their most valuable comments and suggestions which improved the quality of the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
