Some Simplifications in Basic Complex Analysis

Abstract

This paper presents very simple and easy integration-free proofs in the context of Weierstrass's theory of functions, of the Maximum and Minimum Modulus Principles and Gutzmer-Parseval Inequalities for polynomials and for functions developable in complex power series at every point in their domains, as well as a trivial proof of the Open Mapping Theorem, an intuitive version of Liouville's Theorem, an easy proof of Weierstrass's Theorem on Double Series, a modest extension of Schwarz's Lemma, and some other related results. It also presents easy proofs of the P\'olya-Szeg\"o and P. Erd\"os' Anti-Calculus Proposition, a theorem on saddle points by Bak-Ding-Newman, and the well-known Clunie-Jack Lemma.