Some Simplifications in Basic Complex Analysis
arXiv:1207.3553
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 Pólya-Szegö and P. Erdös' Anti-Calculus Proposition, a theorem on saddle points by Bak-Ding-Newman, and the well-known Clunie-Jack Lemma.
28 pages, corrected typos, a revised argument in section 2, result unchanged; a revised argument in section 6, result unchanged; a remark changed in section 7