## 2 The \(\alpha\)-Divergences Induced by a Pair of Strictly Comparable Weighted Means  
### The \((m,n)\)\(\alpha\)-Divergences  
The point of departure for generalizing the \(\alpha\)-divergences is to rewrite Equation (4) for \(\alpha\in\mathbb{R}\backslash\{0,1\}\) as  
\[I_{\alpha}^{+}(p:q)=\frac{1}{\alpha(1-\alpha)}\int(A_{1-\alpha}(p,q)-G_{1- \alpha}(p,q))\mathrm{d}\mu, \tag{16}\]where \(A_{\lambda}\) and \(G_{\lambda}\) for \(\lambda\in(0,1)\) stands for the weighted arithmetic mean and the weighted geometric mean, respectively:  
\[A_{\lambda}(x,y) = (1-\lambda)x+\lambda y,\] \[G_{\lambda}(x,y) = x^{1-\lambda}y^{\lambda}.\]  
For a weighted mean \(M_{\lambda}(a,b)\), we choose the (geometric) convention \(M_{0}(x,y)=x\) and \(M_{1}(x,y)=1\) so that \(\{M_{\lambda}(x,y)\}_{\lambda\in[0,1]}\) smoothly interpolates between \(x\) (\(\lambda=0\)) and \(y\) (\(\lambda=1\)). For the converse convention, we simply define \(M_{\lambda}^{\prime}(a,b)=M_{1-\lambda}(a,b)\) and get the conventional definition of \(I_{\pi}^{+}(p:q)=\frac{1}{\alpha(1-\alpha)}\int(A_{\alpha}^{\prime}(p,q)-G_{ \alpha}^{\prime}(p,q))\mathrm{d}\mu\). In general, a mean \(M(x,y)\) aggregates two values \(x\) and \(y\) of an interval \(I\subset\mathbb{R}\) to produce an intermediate quantity which satisfies the innerness property [35; 36]:  
\[\min\{x,y\}\leq M(x,y)\leq\max\{x,y\},\quad\forall x,y\in I. \tag{17}\]  
This in-between property of means (Equation (17)) was postulated by [PERSON] [37] in 1821. A mean is said strict if the inequalities of Equation (17) are strict whenever \(x\
eq y\). A mean \(M\) is said reflexive iff \(M(x,x)=x\) for all \(x\in I\). The reflexive property of means was postulated by [PERSON] [38] in 1929. In the remainder, we consider \(I=(0,\infty)\). By using the unique dyadic representation of any real \(\lambda\in(0,1)\) (i.e., \(\lambda=\sum_{i=1}^{\infty}\frac{d_{i}}{2^{i}}\) with \(d_{i}\in\{0,1\}\) the binary digit expansion of \(\lambda\)), one can build a weighted mean \(M_{\lambda}\) from any given mean \(M\); see [29] for such a construction. In the remainder, we drop the \"+\" notation to emphasize that the divergences are defined between positive measures. By analogy to the \(\alpha\)-divergences, let us define the (decomposable) \((M,N)\)\(\alpha\)-divergences between two positive densities \(p\) and \(q\) for a pair of weighted means \(M_{1-\alpha}\) and \(N_{1-\alpha}\) for \(\alpha\in(0,1)\) as  
\[I_{\alpha}^{M,N}(p:q):=\frac{1}{\alpha(1-\alpha)}\int(M_{1-\alpha}(p,q)-N_{1- \alpha}(p,q))\mathrm{d}\mu. \tag{18}\]  
The ordinary \(\alpha\)-divergences for \(\alpha\in(0,1)\) are recovered as the \((A,G)\)\(\alpha\)-divergences:  
\[I_{\alpha}^{A,G}(p:q) = \frac{1}{\alpha(1-\alpha)}\int(A_{1-\alpha}(p,q)-G_{1-\alpha}(p,q ))\mathrm{d}\mu, \tag{19}\] \[= I_{1-\alpha}(p:q)=I_{\alpha}(q:p)=I_{\alpha}^{*}(p:q). \tag{20}\]  
In order to define generalized \(\alpha\)-divergences satisfying axioms D1 and D2 of proper divergences, we need to characterize the class of acceptable means. We give a definition strengthening the notion of comparable means in [29]:  
**Definition 1** (Strictly comparable weighted means). : _A pair \((M,N)\) of means are said strictly comparable whenever \(M_{\lambda}(x,y)\geq N_{\lambda}(x,y)\) for all \(x,y\in(0,\infty)\) with equality if and only if \(x=y\), and for all \(\lambda\in(0,1)\)._  
**Example 1**. : _For example, the inequality of the arithmetic and geometric means states that \(A(x,y)\geq G(x,y)\) implies means \(A\) and \(G\) are comparable, denoted by \(A\geq G\). Furthermore, the arithmetic and geometric weighted means are distinct whenever \(x\
eq y\). Indeed, consider the equation \((1-\alpha)x+\alpha y=x^{1-\alpha}y^{\alpha}\) for \(x,y>0\) and \(x\
eq y\). By taking the logarithm on both sides, we get_  
\[\log((1-\alpha)x+\alpha y)=(1-\alpha)\log x+\alpha\log y. \tag{21}\]  
_Since the logarithm is a strictly convex function, the only solution is \(x=y\).