What would you like to understand?

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.

The estimation lemma (ML inequality) is applied by bounding the absolute value of the integral of $f(w) - f(a)$ over the path $\gamma_z$ by the supremum of $|f(w) - f(a)|$ on $\gamma_z$ multiplied by the length of the path $\gamma_z$. Since $\gamma_z$ is the straight line segment from a to z, its length is $|z - a|$.

Conditions: The path $\gamma_z$ is a rectifiable contour (specifically, a straight line segment).; The function f is continuous on $\gamma_z$.; The integrand is $f(w) - f(a)$.

The estimation lemma (ML inequality) is applied by bounding the absolute value of the integral of $f(w) - f(a)$ over the path $\gamma_z$ by the supremum of $|f(w) - f(a)|$ on $\gamma_z$ multiplied by the length of the path $\gamma_z$. Since $\gamma_z$ is the straight line segment from a to z, its length is $|z - a|$.

Conditions: The path $\gamma_z$ is a rectifiable contour (specifically, a straight line segment).; The function f is continuous on $\gamma_z$.; The integrand is $f(w) - f(a)$.

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.