Minimum modulus principle

A nonconstant holomorphic function cannot have a positive interior local minimum of its modulus. Wherever it is nonzero, its reciprocal is holomorphic; a positive modulus minimum would give a modulus maximum of the reciprocal, contradicting the maximum modulus principle. Thus if a holomorphic function on a neighborhood of a closed disk has modulus at the center strictly smaller than its minimum boundary modulus, it has a zero inside the disk. Otherwise the Cauchy estimate applied to its reciprocal would give the opposite inequality.