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

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.