Convexity of Polynomials Using Positivity of Trigonometric Sums

Abstract

Positivity of trigonometric polynomials is of interest for more than a century because of its applications. In this work, we use positivity of trigonometric sine and cosine sums to find the convexity of a polynomial $f(z)=\displaystyle \sum _{k=1}^n a_kz^k$. Further, we also investigate the radius of convexity r such that $f(\mathbb {D}_{\rho })$ is convex where $\mathbb {D}_{\rho }=\{z;|z|\leq \rho ,\, 0<\rho <1\}$.