## 2 The \(\alpha\)-Divergences Induced by a Pair of Strictly Comparable Weighted Means
### Limit Cases of \(1\)-divergences and \(0\)-divergences
Thus, in both cases, we checked that \(E_{f}(p(x),q(x))\geq E_{g}(p(x),q(x))\). Therefore, \(I_{1}^{f,g}(p:q)\geq 0\), and since the QAMs are distinct, \(I_{1}^{f,g}(p:q)=0\) iff \(p(x)=q(x)\)\(\mu\)-a.e. We can interpret the \(I_{1}\) divergences as generalized KL divergences and define generalized notions of cross-entropies and entropies. Since the KL divergence can be written as the cross-entropy minus the entropy, we can also decompose the \(I_{1}\) divergences as follows:  
\[f^{-1}(x)=f^{-1}(x_{0})+(x-x_{0})(f^{-1})^{\prime}(x_{0})+o(x-x_{0}). \tag{48}\]5 \[f^{-1}(x)=f^{-1}(x_{0})+(x-x_{0})(f^{-1})^{\prime}(x_{0})+o(x-x_{0}). \tag{48}\]6  
where \(h_{\times}^{f,g}(p:q)\) denotes the \((f,g)\)-cross-entropy (for a constant \(c\in\mathbb{R}\)):  
\[f^{-1}(x)=f^{-1}(x_{0})+(x-x_{0})(f^{-1})^{\prime}(x_{0})+o(x-x_{0}). \tag{48}\]7  
and \(h^{f,g}(p)\) stands for the \((f,g)\)-entropy (self cross-entropy):  
\[f^{-1}(x)=f^{-1}(x_{0})+(x-x_{0})(f^{-1})^{\prime}(x_{0})+o(x-x_{0}). \tag{48}\]8  
Notice that we recover the Shannon entropy for \(f(x)=x\) and \(g(x)=\log(x)\) with \(f\circ g^{-1})(x)=\exp(x)\) (strictly convex) and \(c=-1\) to annihilate the \(\int\mathrm{d}\mu=1\) term:  
\[f^{-1}(x)=f^{-1}(x_{0})+(x-x_{0})(f^{-1})^{\prime}(x_{0})+o(x-x_{0}). \tag{48}\]9  
We define the generalized \((f,g)\)-Kullback-Leibler divergence or generalized \((f,g)\)-relative entropies:  
\[(f^{-1})^{\prime}(x)=\frac{1}{(f^{\prime}(f^{-1})(x))}, \tag{49}\]0When \(f=f_{A}\) and \(g=f_{G}\), we resolve the constant to \(c=0\), and recover the ordinary Shannon cross-entropy and entropy:  
\[(f^{-1})^{\prime}(x)=\frac{1}{(f^{\prime}(f^{-1})(x))}, \tag{49}\]1 \[(f^{-1})^{\prime}(x)=\frac{1}{(f^{\prime}(f^{-1})(x))}, \tag{49}\]2  
and we have the \((f_{A},f_{G})\)-Kullback-Leibler divergence that is the extended Kullback-Leibler divergence:  
\[(f^{-1})^{\prime}(x)=\frac{1}{(f^{\prime}(f^{-1})(x))}, \tag{49}\]3  
Thus, we have the \((f,g)\)-cross-entropy and \((f,g)\)-entropy expressed as  
\[(f^{-1})^{\prime}(x)=\frac{1}{(f^{\prime}(f^{-1})(x))}, \tag{49}\]4 \[(f^{-1})^{\prime}(x)=\frac{1}{(f^{\prime}(f^{-1})(x))}, \tag{49}\]5  
In general, we can define the \((f,g)\)-Jeffreys divergence as:  
\[(f^{-1})^{\prime}(x)=\frac{1}{(f^{\prime}(f^{-1})(x))}, \tag{49}\]6  
Thus, we define the quasi-arithmetic mean \(\alpha\)-divergences as follows:  
**Theorem 3** (Quasi-arithmetic \(\alpha\)-divergences). : _Let \(f\) and \(g\) be two strictly continuously increasing and differentiable functions on \((0,\infty)\) such that \(f\circ g^{-1}\) is strictly convex. Then, the quasi-arithmetic \(\alpha\)-divergences induced by \((f,g)\) for \(\alpha\in[0,1]\) is_  
\[I_{\alpha}^{f,g}(p:q)=\left\{ \begin{array}{ll}\frac{1}{\alpha(1-\alpha)} \int\Bigl{(}M_{1-\alpha}^{f}(p,q)-M_{1-\alpha}^{g}(p,q)\Bigr{)}\mathrm{d}\mu, &\alpha\in\mathbb{R}\backslash\{0,1\}.\\ I_{1}^{f,g}(p:q)=\int\Bigl{(}\frac{f(q)-f(p)}{f^{\prime}(p)}-\frac{g(q)-g(p)}{ g^{\prime}(p)}\Bigr{)}\mathrm{d}\mu&\alpha=1,\\ I_{0}^{f,g}(p:q)=\int\Bigl{(}\frac{f(p)-f(q)}{f^{\prime}(q)}-\frac{g(p)-g(q)}{ g^{\prime}(q)}\Bigr{)}\mathrm{d}\mu,&\alpha=0.\end{array} \right. \tag{70}\]  
When \(f(u)=f_{A}(u)=u\) (\(M^{f}=A\)) and \(g(u)=f_{G}(u)=\log u\) (\(M^{g}=G\)), we get  
\[I_{1}^{A,G}(p:q)=\int\biggl{(}q-p-p\log\frac{q}{p}\biggr{)}\mathrm{d}\mu= \mathrm{KL}^{+}(p:q)=I_{1}(p:q), \tag{71}\]  
the Kullback-Leibler divergence (KLD) extended to positive densities, and \(I_{0}=\mathrm{KL}^{+}{}^{*}\) the reverse extended KLD. Let \(\mathcal{M}\) denote the class of strictly increasing and differentiable real-valued univariate functions. An interesting question is to study the class of pairs of functions \((f,g)\in\mathcal{M}\times\mathcal{M}\) such that \(I_{1}^{f,g}(p:q)=\mathrm{KL}(p:q)\). This involves solving integral-based functional equations [57].