Contour integrals are bounded by maximum size times path length: $\left|\int_\gamma f\,dz\right|\le ML$.Suggest a slogan
Let \(\gamma\colon[a,b]\to\CC\) be a piecewise \(C^1\) curve with length \(L\coloneqq\int_a^b\abs{\gamma'(t)}\dt\), and let \(f\) be continuous on the image of \(\gamma\). With \(M\coloneqq\max_{z\in\gamma}\abs{f(z)}\), \[ \abs{\int_\gamma f(z)\dz}\le ML. \]
By the definition of the contour integral, \[ \abs{\int_\gamma f(z)\dz}=\abs{\int_a^bf(\gamma(t))\gamma'(t)\dt}\le\int_a^b\abs{f(\gamma(t))}\abs{\gamma'(t)}\dt\le M\int_a^b\abs{\gamma'(t)}\dt=ML. \]