The symmetric versions of Rouché's theorem via \(\bar{\partial}\)-calculus (Q2338817)

Summary: Let \((f,g)\) be a pair of holomorphic functions. In this expositional paper we apply the \(\overline\partial\)-calculus to prove the symmetric version ``\(| f+g | < | f| + | g|\)'' as well as the homotopic version of Rouché's theorem for arbitrary planar compacta \(K\). Using Eilenberg's representation theorem we also give a converse to the homotopic version. Then we derive two analogs of Rouché's theorem for continuous-holomorphic pairs (a symmetric and a nonsymmetric one). One of the rarely presented properties of the non-symmetric version is that in the fundamental boundary hypothesis, \(| f+g| \leq | g|\), equality is allowed.