A property of the trigonometric polynomials \(T_{m,p,q}\) (Q1099351)

The trigonometric polynomial \(T_{m,p,q}\) of degree n (assumed fixed in the following) has real coefficients, only real roots, among which 0 has multiplicity m; it assumes the locally extreme values p and (-1) mq at r and -s resp., where r and s are positive and minimal, \(0<p,q\leq 1\), the extreme values assumed at the remaining 2n-m-1 roots of \((T_{m,p,q})'\) in the fundamental interval (-\(\pi\),\(\pi\) ] being \(\pm 1.\) Let \(x_ 1\) and \(x_ 2\) belong to the same root interval of \(f=T_{m,p,q}\). Let \(| x|\) denote the distance from x to the set \(\{\) 2t\(\pi\) \(| t\in {\mathbb{Z}}\}\). Then \(| x_ 1| <| x_ 2| \Rightarrow | f'(x_ 1)| \leq | f'(x_ 2)|\) with strict inequality except in a few simple cases. The polynomials \(T_{m,p,q}\) turn up in extremal problems, and the property mentioned appears to be of interest here. The method of proof is almost certainly more complicated than necessary for this simple problem but enables one to discover relationships between the various polynomials \(T_{m,p,q}\).