On the trigonometric polynomials of Fejér and Young (Q663101)

Let \[ S_{n}(x)=\sum_{k=1}^{n} \frac{\sin(kx)}{k} \;\;\text{and}\;\; C_{n}(x)=1+\sum_{k=1}^{n}\frac{\cos(kx)}{k} \] be the trigonometric sums of Fejér and Young, respectively. The authors prove the following result: For all natural numbers \(n\geq 2\) and real numbers \(x\in (0,\,\pi)\) one has \[ \frac{C_{n}(x)}{S_{n}(x)}\geq \frac{1}{9}\,\sqrt{15}\,.\tag{(1)} \] The sign of equality holds in (1) if and only if \(n=2\) and \(x=\arccos(-1/4)=1.8234\dots.\) The lower bound in (1) is sharp.