Inequalities

The means $\mathcal{\bar{A}}_r(a)$ and $\mathcal{G}(a)$ are of the form

$\mathcal{\bar{A}}_\phi(a) = \phi^{-1}\left(\sum q \phi(a)\right)$,

where $\phi(x)$ is one of the functions: $x^r$ and $\log x$ and $\phi^{-1}(x)$ the inverse function. In a nutshell, for $\phi(x) = x, \log x$ and $x^r$, $\mathcal{\bar{A}}_\phi$ reduces to $\mathcal{A}$, $\mathcal{G}$ and $\mathcal{\bar{A}}_r$ respectively.

$\S$

In order that

$$\mathcal{\bar{A}}_\psi(a) = \mathcal{\bar{A}}_\chi(a)$$
for all $a$ and $q$, it is necessay and sufficient that
$$\chi=\alpha\psi+\beta$$ ,
where $\alpha$ and $\beta$ are contants and $\alpha\neq0$.

Since $-\phi$ is a linear function of $\phi$, and $-\phi$ increases if $\phi$ decreases, we may always suppose, if we please, that the $\phi$ involved in $\mathcal{\bar{A}}_\phi(x)$ is an increasing function.

$\S$

Suppose that $\phi(x)$ is continuous in the open interval $(0,+\infty)$, and that
$\mathcal{\bar{A}}_\phi(ka) = k \mathcal{\bar{A}}_\phi(a)$
for all positive $a$, $q$, and $k$. Then $\mathcal{\bar{A}}_\phi(a)$ is $\mathcal{\bar{A}}_r(a)$.

$\S$

$\log \mathcal{\bar{A}}_r^r(a) = r \log \mathcal{\bar{A}}_r(a)$ is a convex function of $r$.

$\S$

If $\phi(x)$ is continuous, and there is at least one point of every chord of the curve $y=\phi(x)$, besides the end points of the chord, which lies above or on the curve, then every point of every chord lies above or one the curve, so that $\phi(x)$ is convex.

$\S$

If $\psi$ and $\chi$ are continuous and strictly montonic, and $\chi$ is increasing, then a necessary and sufficient condition that $\mathcal{\bar{A}}_\psi\le \mathcal{\bar{A}}_\chi$ for all $a$ and $q$ is that $\phi=\chi\psi^{-1}$ should be convex.

$\S$

If $\phi(x,y)$ is convex and continuous, then
$\phi\left(\sum qx,\sum qy\right) \le \sum q \phi(x,y)$

$\S$

If $F$ and $G$ are continuous and strictly monotonic, then a necessary and sufficient condition that $\mathcal{A}(ab)$ should be comparable with $\mathcal{\bar{A}}_F(a)\mathcal{\bar{A}}_G(b)$ is that $F^{-1}(x)G^{-1}(y)$ should be a concave or convex function of the two variables $x$ and $y$; in the first case

$$\mathcal{A}(ab) \le \mathcal{\bar{A}}_F(a)\mathcal{\bar{A}}_G(b)$$ , in the second the reverse inequality.