Cauchy Integral Inequality: Definition, Applications & Examples

The Cauchy integral inequality states that for a complex-valued function f(z) that is analytic on a closed curve C and continuous on the interior of C, and for any point z0 inside C, we have: \n\n|f(z0)| ≤ (1/2π) * ∫(C) |f(z)/(z-z0)| |dz|\n\nwhere the integral is taken over the curve C.\n\nIn other words, the absolute value of the function evaluated at a point z0 is bounded by the average value of the function along the curve C, multiplied by the length of the curve C.\n\nThis inequality is a consequence of Cauchy's integral formula and has many applications in complex analysis, including estimates on the maximum modulus principle and the existence of zeros of analytic functions.