@robjohn I might be wrong about this, but I remember discussing this problem with you before: http://mathb.in/61239
The part where I write $\frac{1}{2\pi |\lambda_n|}\int_{-\pi}^\pi f(t) D_n(-t)\, dt \ge \frac{m\|D_n\|_1}{|\lambda_n|}$ isn't correct, since $\|D_n\|_1$ would require $|D_n(-t)|$ in the integral, and we have the same thing without the modulus. Could you please take a look and see how to fix it?