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The factors cancel because the ML inequality introduces the length of the integration path, which is $|z-a|$ for the straight line segment from a to z. This length factor multiplies the supremum term, and it exactly cancels with the $1/|z-a|$ factor that comes from the denominator of the difference quotient.

Conditions: The integration path is the straight line segment from a to z.; The difference quotient has a denominator of $z-a$, whose absolute value is $|z-a|$.; The ML inequality is applied to the integral over this segment.

The factors cancel because the ML inequality introduces the length of the integration path, which is $|z-a|$ for the straight line segment from a to z. This length factor multiplies the supremum term, and it exactly cancels with the $1/|z-a|$ factor that comes from the denominator of the difference quotient.

Conditions: The integration path is the straight line segment from a to z.; The difference quotient has a denominator of $z-a$, whose absolute value is $|z-a|$.; The ML inequality is applied to the integral over this segment.