2012/07/16 by Oswaldo Rio Branco de Oliveira, de Oliveira, Oswaldo Rio Branco · 1 citation
Mathematics · #30-01 #30A10 #30B10 #Analytic Number Theory Research #Complex Variables (math.CV) #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #math.CV #msc:30-01 #msc:30A10 #msc:30B10
paper · pdf · doi:10.48550/arxiv.1207.3553
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
openalex publication_date 2012/07/16 · arxiv created 2012/08/09 · arxiv updated 2012/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.