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A proof of a trigonometric inequality. A glimpse inside the mathematical kitchen

 

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Author: Arias de Reyna Martínez, Juan
Van de Lune, Jan
Department: Universidad de Sevilla. Departamento de Análisis Matemático
Date: 2011-09
Published in: Journal of Mathematical Inequalities, 5(3), 341-353
Document type: Article
Abstract: We prove the inequality ∞ ∑ k=1(−1) k+1 rk cos kφ k+2 < ∞ ∑ k=1 (−1) k+1 rk k+2 for 0 < r 1 and 0 < φ < π . For the case r = 1 we give two proofs. The first one is by means of a general numerical technique (Maximal Slope Principle) for proving inequalities between elementary functions. The second proof is fully analytical. Finally we prove a general rearrangement theorem and apply it to the remaining case 0 < r < 1. Some of these inequalities are needed for obtaining general sharp bounds for the errors committed when applying the Riemann-Siegel expansion of Riemann’s zeta function.
Cite: Arias de Reyna Martínez, J. y Van de Lune, J. (2011). A proof of a trigonometric inequality. A glimpse inside the mathematical kitchen. Journal of Mathematical Inequalities, 5 (3), 341-353.
Size: 209.4Kb
Format: PDF

URI: http://hdl.handle.net/11441/40416

DOI: 10.7153/jmi-05-30

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