We are given a positive integer n and real numbers a_{i} such that |∑_{1}^{n} a_{k} sin kx| ≤ |sin x| for all real x. Prove |∑_{1}^{n} k a_{k}| ≤ 1.

**Solution**

Put f(x) = ∑_{1}^{n} a_{k} sin kx. We note that ∑_{1}^{n} k a_{k} = f '(0).

We also have f '(0) = lim ( f(x) - f(0) )/x = lim f(x)/x = lim f(x)/sin x lim (sin x)/x = lim f(x)/sin x. But |f(x)| ≤ |sin x|, so |f(x)/sin x| ≤ 1 and hence |f '(0)| = |lim f(x)/sin x| ≤ 1.

© John Scholes

jscholes@kalva.demon.co.uk

14 Jan 2002