The generalized mean $M$ (also called power mean or Hölder mean) of two numbers $a$ and $b$ of degree $p$ is
$$
M_{p}(a,b) = \left(\frac{a^{p} + b^{p}}{2}\right)^{\frac{1}{p}}
$$
The following derivations often make use of the trick
$$
a^x = e^{\ln a^x} = e^{x\ln a}
$$
and also use the rule of L'Hopital
$$
\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}
$$
The arithemtic mean follows from the generalized mean for $p=1$:
$$
M_{1}(a,b) = \frac{a + b}{2}
$$
The harmonic mean is the particular case of $p=-1$:
$$
M_{-1}(a,b) = \left(\frac{\frac{1}{a} + \frac{1}{b}}{2}\right)^{-1} = \frac{2}{\frac{1}{a} + \frac{1}{b}} = \frac{2ab}{a+b}
$$
The root mean square follows from the generalized mean for $p=2$:
$$
M_{2}(a,b) = \sqrt{\frac{a^2 + b^2}{2}}
$$
The rarely used harmonic mean of squares is the special case for $p=-2$:
$$
M_{-2}(a,b) = \left(\frac{\frac{1}{a^2} + \frac{1}{b^2}}{2}\right)^{-1/2} = \sqrt{\frac{2}{\frac{1}{a^2} + \frac{1}{b^2}}} = \sqrt{\frac{2a^2b^2}{a^2+b^2}}
$$
The geometric mean is the generalized mean in the limit $p \to 0$ :
$$
\begin{align}
M_0(a,b) &= \lim_{p \to 0} \left(\frac{a^p + b^p}{2}\right)^{\frac{1}{p}} = \lim_{p \to 0} e^{\ln \left(\frac{a^p + b^p}{2}\right)^{\frac{1}{p}} } = \lim_{p \to 0} e^{\frac{\ln (a^p + b^p) -\ln 2}{p}} \\
&= \exp\left(\lim_{p \to 0} \frac{ \frac{d}{dp}\left(\ln (a^p + b^p) -\ln 2\right)}{\frac{d}{dp} p}\right) = \exp\left(\lim_{p \to 0} \frac{ \frac{a^p \ln a + b^p \ln b}{a^p+b^p}}{1}\right) \\
&= e^{\frac{1}{2} (\ln a + \ln b)} = e^{\ln (ab)^\frac{1}{2} } \\
&= \sqrt{ab}
\end{align}
$$
In the limit $p \to \infty$ the generalized mean is actually computing the maximum of two numbers $a$ and $b$. To see this we start from the point after we applied the rule of L'Hopital in the derivation for the geometric mean
$$
\begin{align}
M_\infty(a,b) &= \exp\left(\lim_{p \to \infty} \frac{ \frac{a^p \ln a + b^p \ln b}{a^p+b^p}}{1}\right) = \exp\left(\lim_{p \to \infty} \frac{a^p \ln a}{a^p+b^p} + \frac{b^p \ln b}{a^p+b^p}\right) \\
&= \exp\left(\lim_{p \to \infty} \frac{\ln a}{1+\left(\frac{b}{a}\right)^p} + \frac{\ln b}{\left(\frac{a}{b}\right)^p+1}\right)
\end{align}
$$
Now if $a > b$ it means that $\frac{a}{b} > 1$ and $\frac{b}{a} < 1$. From this it follows that $\left(\frac{a}{b}\right)^\infty = \infty$ and $\left(\frac{b}{a}\right)^\infty = 0$ and vice versa for $b > a$. So we get
$$
M_\infty(a,b) =
\left.
\begin{cases}
\exp\left(\frac{\ln a}{1+0} + \frac{\ln b}{\infty+1}\right) = e^{\ln a} = a & \text{for} & a > b \\
\exp\left(\frac{\ln a}{1+\infty} + \frac{\ln b}{0+1}\right) = e^{\ln b} = b & \text{for} & a < b
\end{cases}
\right\} = \max(a,b)
$$
In the limit $p \to -\infty$ the generalized mean is computing the minimum of two numbers $a$ and $b$. The argument follows the derivation for the maximum.
If $a > b$ it means that $\frac{a}{b} > 1$ and $\frac{b}{a} < 1$. From this it follows that $\left(\frac{a}{b}\right)^{-\infty} = \left(\frac{b}{a}\right)^{\infty} = 0$ and $\left(\frac{b}{a}\right)^{-\infty} = \left(\frac{a}{b}\right)^{\infty} = \infty$ and vice versa for $b > a$. So we get
$$
M_{-\infty}(a,b) =
\left.
\begin{cases}
\exp\left(\frac{\ln a}{1+\infty} + \frac{\ln b}{0+1}\right) = e^{\ln b} = b & \text{for} & a > b \\
\exp\left(\frac{\ln a}{1+0} + \frac{\ln b}{0+\infty}\right) = e^{\ln a} = a & \text{for} & a < b
\end{cases}
\right\} = \min(a,b)
$$
An even more general mean is the generalized f-mean $M$ (also called Kolmogorov mean) of two numbers $a$ and $b$ of degree $p$
$$
M_{f}(a,b) = f^{-1}\left(\frac{f(a) + f(b)}{2}\right)
$$
where $f^{-1}(x)$ is the inverse function of $f(x)$ with the property $f^{-1}(f(x)) = x$.1
Using this formulation it is easier to express the harmonic mean and the geometric mean.
The harmonic mean follows from the generalized f-mean for $f=\frac{1}{x}$:2
$$
M_{1/x}(a,b) = \left(\frac{\frac{1}{a} + \frac{1}{b}}{2}\right)^{-1} = \frac{2}{\frac{1}{a} + \frac{1}{b}}
$$
The geometric mean is the particular case of $f=\ln x$ of the generalized f-mean
$$
M_{\ln x}(a,b) = e^{\frac{1}{2} (\ln a + \ln b)} = e^{\ln (ab)^\frac{1}{2} } = \sqrt{ab}
$$
Our familiar power mean/generalized mean as defined above corresponds to the generalized f-mean for $f(x) = x^p$
$$
M_{x^p}(a,b) = \left(\frac{a^{p} + b^{p}}{2}\right)^{\frac{1}{p}}
$$
Footnotes
1 That means the inverse function for $f(x) = x$ is $x$ and not $\frac{1}{x}$. The ladder would be the so called multiplicative inverse with the property $(f(x))^{-1} f(x) = 1$. Don't confuse the multiplicative inverse with the inverse function $f^{-1}(x)$ used in the definition of the generalized f-mean. ⏎
2 The inverse function of $\frac{1}{x}$ is $f^{-1}(x) = \frac{1}{x}$, because $f^{-1}(f(x)) = \frac{1}{\frac{1}{x}} = x$. ⏎
