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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 $f(w) - f(a)$.

We can use a straight line segment because the function f is holomorphic on a simply connected domain D, which guarantees that the contour integral of f is path-independent. This allows us to strategically choose the simplest path—the straight line segment from a to z—to evaluate the difference $F(z) - F(a)$.

Conditions: The function f is holomorphic on the domain D.; The domain D is simply connected.; The points a and z are in D, and the straight line segment connecting them lies entirely within D.

We can use a straight line segment because the function f is holomorphic on a simply connected domain D, which guarantees that the contour integral of f is path-independent. This allows us to strategically choose the simplest path—the straight line segment from a to z—to evaluate the difference $F(z) - F(a)$.

Conditions: The function f is holomorphic on the domain D.; The domain D is simply connected.; The points a and z are in D, and the straight line segment connecting them lies entirely within D.

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.

The notation $[a,z]$ represents the straight line segment joining the points a and z in the complex plane. It is used both as the path of integration for the contour integral and as the set over which the supremum of the function's deviation is taken.

Conditions: The points a and z are in the domain D.; The straight line segment connecting a and z lies entirely within D.

The notation $[a,z]$ represents the straight line segment joining the points a and z in the complex plane. It is used both as the path of integration for the contour integral and as the set over which the supremum of the function's deviation is taken.

Conditions: The points a and z are in the domain D.; The straight line segment connecting a and z lies entirely within D.