## 2 The \(\alpha\)-Divergences Induced by a Pair of Strictly Comparable Weighted Means
### The Quasi-Arithmetic \(\alpha\)-Divergences  
A quasi-arithmetic mean (QAM) is defined for a continuous and strictly monotonic function \(f:I\subset\mathbb{R}_{+}\to J\subset\mathbb{R}_{+}\) as:  
\[M^{f}(x,y):=f^{-1}\bigg{(}\frac{f(x)+f(y)}{2}\bigg{)}. \tag{34}\]  
Function \(f\) is called the generator of the quasi-arithmetic mean. These strict and reflexive quasi-arithmetic means are also called Kolmogorov means [30], Nagumo means [42] de [PERSON] means [43], or quasi-linear means [44] in the literature. These means are called quasi-arithmetic means because they can be interpreted as arithmetic means on the arguments \(f(x)\) and \(f(y)\):  
\[f(M^{f}(x,y))=\frac{f(x)+f(y)}{2}=A(f(x),f(y)). \tag{35}\]  
QAMs are strict, reflexive, and symmetric means. Without loss of generality, we may assume strictly increasing functions \(f\) instead of monotonic functions since \(M^{-f}=M^{f}\). Indeed, \(M^{-f}(x,y)=(-f)^{-1}(-f(M^{f}(x,y)))\) and \(((-f)^{-1}\circ(-f))(u)=u\), the identity function. Notice that the composition \(f_{1}\circ f_{2}\) of two strictly monotonic increasing functions \(f_{1}\) and \(f_{2}\) is a strictly monotonic increasing function. Furthermore, we consider \(I=J=(0,\infty)\) in the remainder since we apply these means on positive densities. Two quasi-arithmetic means \(M^{f}\) and \(M^{g}\) coincide if and only if \(f(u)=ag(u)+b\) for some \(a>0\) and \(b\in\mathbb{R}\), see [44]. The quasi-arithmetic means were considered in the axiomatization of the entropies by [PERSON] to define the \(\alpha\)-entropies (see Equation (2).11 of [45]). By choosing \(f_{A}(u)=u\), \(f_{G}(u)=\log u\), or \(f_{H}(u)=\frac{1}{u}\), we obtain the Pythagorean's arithmetic \(A\), geometric \(G\), and harmonic \(H\) means, respectively:  
* the arithmetic mean (A): \(A(x,y)=\frac{x+y}{2}=M^{f_{A}}(x,y)\),
* the geometric mean (G): \(G(x,y)=\sqrt{xy}=M^{f_{G}}(x,y)\), and
* the harmonic mean (H): \(H(x,y)=\frac{2}{\frac{1}{1+\frac{1}{2}}}=\frac{2 xy}{x+y}=M^{f_{H}}(x,y)\). More generally, choosing \(f_{P_{r}}(u)=u^{r}\), we obtain the parametric family of power means also called Holder means [46] or binary means [47]:  
\[P_{r}(x,y)=\left(\frac{x^{r}+y^{r}}{2}\right)^{\frac{1}{r}}=M^{f_{P_{r}}}(x,y), \quad r\in\mathbb{R}\backslash\{0\}. \tag{36}\]  
In order to get a smooth family of power means, we define the geometric mean as the limit case of \(r\to 0\):  
\[P_{0}(x,y)=\lim_{r\to 0}P_{r}(x,y)=G(x,y)=\sqrt{xy}. \tag{37}\]  
A mean \(M\) is positively homogeneous if and only if \(M(ta,tb)=t\,M(a,b)\) for any \(t>0\). It is known that the only positively homogeneous quasi-arithmetic means coincide exactly with the family of power means [44]. The weighted QAMs are given by  
\[M^{f}_{\alpha}(p,q) = f^{-1}((1-\alpha)f(p)+\alpha f(q))), \tag{38}\] \[= f^{-1}(f(p)+\alpha(f(q)-f(p)))=M^{f}_{1-\alpha}(q,p). \tag{39}\]  
Let us remark that QAMs were generalized to complex-valued generators in [48] and to probability measures defined on a compact support in [49]. Notice that there exist other positively homogeneous means which are not quasi-arithmetic means. For example, the logarithmic mean [50; 51]\(L(x,y)\) for \(x>0\) and \(y>0\):  
\[L(x,y)=\frac{y-x}{\log y-\log x} \tag{40}\]  
is an example of a homogeneous mean (i.e., \(L(tx,ty)=t\,L(x,y)\) for any \(t>0\)) that is not a QAM. Besides the family of QAMs, there exist many other families of means [35]. For example, let us mention the Lagrangian means [52], which intersect with the QAMs only for the arithmetic mean, or a generalization of the QAMs called the Bajraktarevic means [53]. Let us now strengthen a recent theorem (Theorem 1 of [54], 2010):  
**Theorem 1** (Strictly comparable weighted QAMs). : _The pair \((M^{f},M\!s)\) of quasi-arithmetic means obtained for two strictly increasing generators \(f\) and \(g\) is strictly comparable provided that function \(f\circ g^{-1}\) is strictly convex, where \(\circ\) denotes the function composition._  
Proof.