Relationship between the derivative of $f(x)/x$ and the second derivative of $f(x)$ In economics, a "progressive tax" is a tax in which the average tax rate (taxes paid / personal income) increases as the taxable amount increases.
This can be formally described as having a tax function $T$ of income $z$ such as $\frac{\partial T(z)/z}{\partial z} > 0$ (definition 1).
An alternative definition of a progressive tax is that marginal rates are increasing, that is $\frac{\partial^2 T}{\partial z^2} > 0$ (definition 2).
The second definition implies the first definition.
It is not clear (to me, at least) under what assumptions on $T$ does the first definition implies the second.
It is not true for any $T$. For instance if $T$ is defined as $T(z) = a \times z - k$, with $a > 0$ and $k>0$, then the marginal rate is $a$ (constant, and not increasing) and the average rate is $a - k/z$ (increasing).
The question is then: when does the first definition implies the second one?
 A: \begin{align*}
\frac{\mathrm{d}}{\mathrm{d}z} \frac{T(z)}{z} 
&= \frac{T'(z)}{z} - \frac{T(z)}{z^2}  \\
\frac{\mathrm{d}^2}{\mathrm{d}z^2} \frac{T(z)}{z} 
&= \frac{-2}{z}\left( \frac{T'(z)}{z} - \frac{T(z)}{z^2} \right) + \frac{T''(z)}{z}
\end{align*}
When we assume the first is positive, we assume the parenthesized term in the second is positive.  In particular, solving for that term and applying the assumed inequality,
$$  -z\left( \frac{\mathrm{d}^2}{\mathrm{d}z^2} \frac{T(z)}{z} \right) + T''(z) > 0  \text{.}  $$
So the first definition implies $T''(z) > z \frac{\mathrm{d}^2}{\mathrm{d}z^2} \frac{T(z)}{z}$.  Normally, we assume $z$, the taxable amount, is positive, so as long as $\frac{\mathrm{d}^2}{\mathrm{d}z^2} \frac{T(z)}{z} > 0$, the second definition follows.  
By replacing $\frac{\mathrm{d}^2}{\mathrm{d}z^2} \frac{T(z)}{z} > 0$ with $\frac{\mathrm{d}^2}{\mathrm{d}z^2} \frac{T(z)}{z} = 0$ and solving the resulting differential equation, the frontier for this condition is $T(z) = c_2 z^2 + c_1 z$, where $c_1$ and $c_2$ are arbitrary constants.  When $T$ is of this form, it satisfies both definitions if $c_2 > 0$ and satisfies neither definition if $c_2 \leq 0$.  This is compatible with your observation about $T$ linear in $z$, where $c_2 = 0$.
