An exact inequality for the derivative of a trigonometric polynomial having only real zeros (Q1821286)

The following theorem is proved: If a trigonometric polynomial \(T_ n\) has exactly 2n zeros (in \([-\pi,\pi]\)) then \(\| T_ n'\| \geq n/2\cdot (1-1/2n)^{n-}\cdot \| T_ n\|\) and, moreover, the above inequality becomes an equality only for the polynomials of the form \(T_ n(x)=a\cdot (\sin (x-\gamma)/2)^{2n}.\)