Definition of minimum-phase system
one thing about a non-minimum phase system (with a rational transfer function), is that it can be thought of as the series concatenation (or cascade) of a minimum-phase system, having identical magnitude response as the given non-min-phase filter, with an all-pass filter.  the APF will have a poles that cancels specific zeros of the min-phase system that are inside the unit circle, and puts in new zeros ("reflected zeros") outside the unit circle.  so the new system always has more phase shift than the min-phase system.

so here's another definition of a minimum phase system:

An LTI system or filter

$$ H(f) = |H(f)| e^{j \arg\{H(f)\}} = |H(f)| e^{j \phi(f)} $$

is minimum phase if and only if the natural phase response, in radians, is the negative of the Hilbert transform of the natural logarithm of the magnitude response:

$$ \phi(f) \triangleq \arg\{ H(f) \} = -\mathscr{H}\big\{ \ln( |H(f)| ) \big\} $$

since

$$\begin{align}
 H(f) &amp; = |H(f)| e^{j \phi(f)} \\
  &amp; = e^{\ln(|H(f)|)} e^{j \phi(f)} \\
  &amp; = e^{\ln(|H(f)|) + j \phi(f)} \\
  &amp; = e^{\ln(H(f))} \\
\end{align}$$

this is relating the real and imaginary parts of the complex natural logarithm of the frequency response. 

$$ \Im\{ \ln(H(f)) \} = -\mathscr{H}\big\{ \Re\{\ln(H(f)) \} \big\} $$

so a question to ask, why is the pole/zero definition stated in the question equivalent to this Hilbert transform definition??