Article
A proof of a trigonometric inequality. A glimpse inside the mathematical kitchen
Author/s | Arias de Reyna Martínez, Juan
Lune, Jan van de |
Department | Universidad de Sevilla. Departamento de Análisis Matemático |
Publication Date | 2011-09 |
Deposit Date | 2016-04-25 |
Published in |
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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 ... 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. |
Funding agencies | Ministerio de Ciencia e Innovación (MICIN). España |
Project ID. | MTM2009-08934 |
Citation | 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. |
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