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.
The ML inequality is an estimation tool for contour integrals. It states that the absolute value of the integral of a function along a path is bounded by the supremum (maximum) of the absolute value of the function on that path, multiplied by the length of the path.
Conditions: The function is integrated along a rectifiable path.; The function is continuous on the path.
The ML inequality is an estimation tool for contour integrals. It states that the absolute value of the integral of a function along a path is bounded by the supremum (maximum) of the absolute value of the function on that path, multiplied by the length of the path.
Conditions: The function is integrated along a rectifiable path.; The function is continuous on the path.