## 5 Conclusions and Discussion  
For two comparable strict means [35]\(M(p,q)\geq N(p,q)\) (with equality holding if and only if \(p=q\)), one can define their \((M,N)\)-divergence as  
\[I^{M,N}(p:q):=4\,\int(M(p,q)-N(p,q))\mathrm{d}\mu. \tag{114}\]  
When the property of strict comparable means extend to their induced weighted means \(M_{\alpha}(p,q)\) and \(N_{\alpha}(p,q)\) (i.e., \(M_{\alpha}(p,q)\geq N_{\alpha}(p,q)\)), one can further define the family of \((M,N)\)\(\alpha\)-divergences for \(\alpha\in(0,1)\):  
\[I_{\alpha}^{M,N}(p:q):=\frac{1}{\alpha(1-\alpha)}\int(M_{1-\alpha}(p,q)-N_{1- \alpha}(p,q))\mathrm{d}\mu, \tag{115}\]  
so that \(I^{M,N}(p:q)=I_{\frac{1}{2}}^{M,N}(p:q)\). When the weighted means are symmetric, the reference duality holds (i.e., \(I_{\alpha}^{M,N}(q:p)=I_{1-\alpha}^{M,N}(p:q)\)), and we can define the \((M,N)\)-equivalent of the Kullback-Leibler divergence, i.e., the \((M,N)\)\(1\)-divergence, as the limit case (when it exists): \(I_{1}^{M,N}(p:q)=\lim_{\alpha\to 1}I_{\alpha}^{M,N}(p:q)\). Similarly, the \((M,N)\)-equivalent of the reverse Kullback-Leibler divergence is obtained as \(I_{0}^{M,N}(p:q)=\lim_{\alpha\to 0}I_{\alpha}^{M,N}(p:q)\). We proved that the quasi-arithmetic weighted means [30]\(M_{\alpha}^{\text{\tiny{$\alpha$}}}\) and \(M_{\alpha}^{\text{\tiny{$\alpha$}}}\) were strictly comparable whenever \(f\circ g^{-1}\) was strictly convex. In the limit cases of \(\alpha\to 0\) and \(\alpha\to 1\), we reported a closed-form formula for the equivalent of the forward and the reverse Kullback-Leibler divergences. We reported closed-form formulas for the quasi-arithmetic \(\alpha\)-divergences \(I_{\alpha}^{f,\text{\tiny{$\alpha$}}}(p:q):=I_{\alpha}^{M,M^{\text{\tiny{$ \alpha$}}}}(p:q)\) for \(\alpha\in[0,1]\) (Theorem 3) and for the subfamily of homogeneous \((r,s)\)-power \(\alpha\)-divergences \(I_{\alpha}^{rs}(p:q):=I_{\alpha}^{\text{\tiny{MPow}}_{rs},\text{\tiny{MPow}}_{ s}}(p:q)\) induced by power means (Corollary 2). The ordinary \((A,G)\)\(\alpha\)-divergences [12], the \((A,H)\)\(\alpha\)-divergences, and the \((G,H)\)\(\alpha\)-divergences are examples of \((r,s)\)-power \(\alpha\)-divergences obtained for \((r,s)=(1,0)\), \((r,s)=(1,-1)\) and \((r,s)=(0,-1)\), respectively. Generalized \(\alpha\)-divergences may prove useful in reporting a closed-form formula between densities of a parametric family \(\{p_{\theta}\}\). For example, consider the ordinary \(\alpha\)-divergences between two scale Cauchy densities \(p_{1}(x)=\frac{1}{\pi}\frac{s_{1}}{x^{2}+s_{1}^{2}}\) and \(p_{2}(x)=\frac{1}{\pi}\frac{s_{2}}{x^{2}+s_{2}^{2}}\); there is no obvious closed-form for the ordinary \(\alpha\)-divergences,but we can report a closed-form for the \((A,H)\)\(\alpha\)-divergences following the calculus reported in [41]:  
\[I_{\alpha}^{A,H}(p_{1}:p_{2}) = \frac{1}{\alpha(1-\alpha)}\bigg{(}1-\int H_{1-\alpha}(p_{1}(x),p _{2}(x))\text{d}\mu(x)\bigg{)}, \tag{116}\] \[= \frac{1}{\alpha(1-\alpha)}\bigg{(}1-\frac{s_{1}s_{2}}{(\alpha s _{1}+(1-\alpha)s_{2})s_{1-\alpha}}\bigg{)}, \tag{117}\]  
with \(s_{\alpha}=\sqrt{\frac{\alpha s_{1}s_{2}^{2}+(1-\alpha)s_{2}s_{1}^{2}}{ \alpha s_{1}+(1-\alpha)s_{2}}}\). For probability distributions \(p_{\theta_{1}}\) and \(p_{\theta_{2}}\) belonging to the same exponential family [77] with cumulant function \(F\), the ordinary \(\alpha\)-divergences admit the following closed-form solution:  
\[I_{\alpha}(p_{\theta_{1}}:p_{\theta_{2}})=\] \[\left\{ \begin{array}{ll}\frac{1}{\alpha(1-\alpha)}(1-\exp(F( \alpha\theta_{1}+(1-\alpha)\theta_{2})-(\alpha F(\theta_{1})+(1-\alpha)F( \theta_{2}))),&\alpha\in(0,1)\\ I_{1}(p_{\theta_{1}}:p_{\theta_{2}})=\text{KL}(p_{\theta_{1}}:p_{\theta_{2}})= B_{F}(\theta_{2}:\theta_{1}),&\alpha=1\\ I_{0}(p_{\theta_{1}}:p_{\theta_{2}})=\text{KL}(p_{\theta_{2}}:p_{\theta_{1}})= B_{F}(\theta_{1}:\theta_{2})&\alpha=0\end{array} \right. \tag{118}\]  
where \(B_{F}\) is the Bregman divergence: \(B_{F}(\theta_{2}:\theta_{1})=F(\theta_{2})-F(\theta_{1})-(\theta_{2}-\theta_{1 })^{\top}\
abla F(\theta_{1})\).