Estimation lemma

The estimation lemma bounds a complex contour integral by multiplying the curve’s length by an upper bound for the integrand’s modulus along the curve.

Contour integral: A complex integral taken along an oriented curve in the complex plane. The lemma bounds this integral by controlling the integrand and the curve’s length.
Cauchy integral formula: A formula expressing the value of a holomorphic function inside a contour through an integral over that contour. The estimation lemma bounds the formula’s contour integral in applications.
Complex-valued function: A function whose values are complex numbers. The lemma applies a modulus bound to function values along a curve.
ML inequality for a circular arc: An application of the estimation lemma using a circular arc’s radius and subtended angle to compute its length. It makes the length factor explicit as radius times angle.
Contour integral parametrization: A representation of a contour integral as an ordinary integral over a curve parameter. Parametrization turns curve length into the factor that appears in the bound.
Cauchy estimates: Bounds on derivatives of a holomorphic function obtained from the Cauchy integral formula. The lemma supplies the integral bound used to derive derivative estimates.
Piecewise smooth curve: A curve made of finitely many smooth segments, suitable for standard contour integration. Its finite length and regular parametrization make the estimate applicable.
ML inequality for a line segment: An application of the estimation lemma to an integral over a straight segment of finite length. The path length is simply the distance between the segment’s endpoints.
Triangle inequality: The inequality stating that the magnitude of a sum is no greater than the sum of the magnitudes. Its integral form bounds the modulus of an integral by the integral of the modulus.
Liouville's theorem: The theorem that every bounded entire function is constant. Contour estimates help bound derivatives and establish this rigidity result.

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