A NEW REFINEMENT OF YOUNG'S INEQUALITY

Abstract

A classical theorem due to Young states that the cosine polynomial

$$ C_n(x)=1+\sum_{k=1}^{n}\frac{\cos(kx)}{k} $$

is positive for all $n\geq1$ and $x\in(0,\pi)$. We prove the following refinement. For all $n\geq2$ and $x\in[0,\pi]$ we have

$$ \tfrac{1}{6}+c(\pi-x)^2\leq C_n(x), $$

with the best possible constant factor

$$ c=\min_{0\leq t\lt\pi}\frac{5+6\cos(t)+3\cos(2t)}{6(\pi-t)^2}=0.069\dots. $$