Contour integrals are bounded by maximum size times path length.Suggest a slogan
Let \(\gamma\colon[a,b]\to\CC\) be a piecewise \(C^1\) curve with length \(\ell(\gamma)\coloneqq\int_a^b\abs{\gamma'(t)}\dt\), and let \(f\) be continuous on \(\gamma([a,b])\). Then \[ \abs{\int_\gamma f\dz}\le\sup_{z\in\gamma([a,b])}\abs{f(z)}\cdot\ell(\gamma). \]
By the definition of the contour integral, \[ \abs{\int_\gamma f\dz}=\abs{\int_a^bf(\gamma(t))\gamma'(t)\dt}\le\int_a^b\abs{f(\gamma(t))}\,\abs{\gamma'(t)}\dt\le\sup_{z\in\gamma([a,b])}\abs{f(z)}\int_a^b\abs{\gamma'(t)}\dt. \]
State the maximum length (ML) estimate for a contour integral.