On the quasi-convexity of trinomial arcs (Q1373421)

For a given integer \(m\geq 3\), define the function \(\theta\mapsto\rho(\theta)\) implicitly by \[ \rho\cdot\sin m\theta- \rho^m\sin \theta= \sin(m-1)\theta. \] By simple arguments from calculus, the authors prove: \(\theta\mapsto \rho(\theta)\) is increasing in \({\pi\over m-1}<\theta< {2\pi\over m}\), and decrasing in \(\pi- {\pi\over m}<\theta< \pi\), and they obtain some results for the function \(\rho^*(\theta)\), defined by \[ (\rho^*)^q\cdot\sin s\theta- (\rho^*)^s\cdot\sin q\theta= \sin(s- q)\theta. \]