## 3 The Subfamily of Homogeneous \((r,s)\)-Power \(\alpha\)-divergences for \(r>s\)  
In particular, we can define the \((r,s)\)-power \(\alpha\)-divergences from two power means \(P_{r}=M^{\text{pow}_{r}}\) and \(P_{s}=M^{\text{pow}_{s}}\) with \(r>s\) (and \(P_{r}\geq P_{s}\)) with the family of generators \(\text{pow}_{l}(u)=u^{l}\). Indeed, we check that \(f_{rs}(u):=\text{pow}_{r}\circ\text{pow}_{s}^{-1}(u)=u^{\frac{r}{s}}\) is strictly convex on \((0,\infty)\) since \(f^{\prime\prime}_{rs}(u)=\frac{r}{s}\big{(}\frac{r}{s}-1\big{)}u^{\frac{r}{s}- 2}>0\) for \(r>s\). Thus, \(P_{r}\) and \(P_{s}\) are two QAMs which are both comparable and distinct. Table 1 lists the expressions of \(E_{r}(p,q):=E_{\text{pow}_{r}}(p,q)\) obtained from the power mean generators \(\text{pow}_{r}(u)=u^{r}\). We conclude with the definition of the \((r,s)\)-power \(\alpha\)-divergences:  
**Corollary 2** (power \(\alpha\)-divergences). : _Given \(r>s\), the \(\alpha\)-power divergences are defined for \(r>s\) and \(r,s\
eq 0\) by_  
\[l_{\alpha}^{rs}(p:q)=\left\{ \begin{array}{ll}\frac{1}{\alpha(1-\alpha)}\int \Big{(}(\alpha p^{r}+(1-\alpha)q^{r})^{\frac{1}{s}}-(\alpha p^{s}+(1-\alpha)q ^{s})^{\frac{1}{s}}\Big{)}\mathrm{d}\mu,&\alpha\in\mathbb{R}\backslash\{0,1\}.\\ \par I_{1}^{rs}(p:q)=\int\Big{(}\frac{q^{r}-p^{r}}{p^{r}-1}-\frac{q^{r}-p^{s}}{ sp^{r}-1}\Big{)}\mathrm{d}\mu&\alpha=1,\\ \par I_{0}^{rs}(p:q)=I_{1}^{rs}(q:p)&\alpha=0.\end{array} \right. \tag{99}\]  
Figure 1: Interplay of divergences and their information-geometric structures: Bregman divergences are canonical divergences of dually flat structures, and the \(\alpha\)-logarithmic divergences are canonical divergences of 1-conformally flat statistical manifolds. When \(\alpha\to 0\), the logarithmic divergence \(L_{\mathcal{FA}}\) tends to the Bregman divergence \(B_{F}\). When \(r=0\), we get the following power \(\alpha\)-divergences for \(s<0\):  
\[I_{\alpha}^{0,s}(p:q)=\left\{ \begin{array}{ll}\frac{1}{\alpha(1-\alpha)}\int \left(p^{\alpha}q^{1-\alpha}-(\alpha p^{s}+(1-\alpha)q^{s})^{\frac{1}{s}} \right)\mathrm{d}\mu,&\alpha\in\mathbb{R}\backslash\{0,1\}.\\ I_{1}^{0,s}(p:q)=\int\left(p\log\frac{q}{p}-\frac{q^{s}-p^{s}}{sp^{s-1}} \right)\mathrm{d}\mu&\alpha=1,\\ I_{0}^{0,s}(p:q)=I_{1}^{r,s}(q:p)&\alpha=0.\end{array} \right. \tag{100}\]  
When \(s=0\), we get the following power \(\alpha\)-divergences for \(r>0\):  
\[I_{\alpha}^{r,0}(p:q)=\left\{ \begin{array}{ll}\frac{1}{\alpha(1-\alpha)}\int \left((\alpha p^{r}+(1-\alpha)q^{r})^{\frac{1}{s}}-p^{\alpha}q^{1-\alpha} \right)\mathrm{d}\mu,&\alpha\in\mathbb{R}\backslash\{0,1\}.\\ I_{1}^{r,0}(p:q)=\int\left(\frac{q^{r}-p^{s}}{rp^{r-1}}-p\log\frac{q}{p} \right)\mathrm{d}\mu&\alpha=1,\\ I_{0}^{r,0}(p:q)=I_{1}^{r,s}(q:p)&\alpha=0.\end{array} \right. \tag{101}\]  
In particular, we get the following family of \((A,H)\)\(\alpha\)-divergences  
\[I_{\alpha}^{A,H}(p:q)=I_{\alpha}^{1,-1}(p:q)=\left\{ \begin{array}{ll}\frac{1 }{\alpha(1-\alpha)}\int\left(\alpha p+(1-\alpha)q-\frac{pq}{\alpha q+(1-\alpha )p}\right)\mathrm{d}\mu,&\alpha\in\mathbb{R}\backslash\{0,1\}.\\ I_{1}^{1,-1}(p:q)=\int\left(q-2p+\frac{p^{2}}{q}\right)\mathrm{d}\mu&\alpha=1,\\ I_{0}^{1,-1}(p:q)=I_{1}^{1,-1}(q:p)&\alpha=0.\end{array} \right., \tag{102}\]  
and the family of \((G,H)\)\(\alpha\)-divergences:  
\[I_{\alpha}^{G,H}(p:q)=I_{\alpha}^{0,-1}(p:q)=\left\{ \begin{array}{ll}\frac{ 1}{\alpha(1-\alpha)}\int\left(p^{\alpha}q^{1-\alpha}-\frac{pq}{\alpha q+(1- \alpha)p}\right)\mathrm{d}\mu,&\alpha\in\mathbb{R}\backslash\{0,1\}.\\ I_{1}^{0,-1}(p:q)=\int\left(p\log\frac{q}{p}-p+\frac{p^{2}}{q}\right)\mathrm{d} \mu&\alpha=1,\\ I_{0}^{0,-1}(p:q)=I_{1}^{0,-1}(q:p)&\alpha=0.\end{array} \right. \tag{103}\]  
The \((r,s)\)-power \(\alpha\)-divergences for \(r,s\
eq 0\) yield homogeneous divergences: \(I_{\alpha}^{r,s}(tp:tq)=t\,I_{\alpha}^{r,s}(p:q)\) for any \(t>0\) because the power means are homogeneous: \(P_{\alpha}^{r}(tx,ty)=tP_{\alpha}^{r}(x,y)=txP_{\alpha}^{r}\big{(}1,\frac{y}{x} \big{)}\).