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young's inequality

Proceedings of the Edinburgh Mathematical Society (1)

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A NEW REFINEMENT OF YOUNG'S INEQUALITY
Horst Alzer, Stamatis Koumandos
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 50 / Issue 2 / June 2007

Published online by Cambridge University Press:

17 May 2007, pp. 255-262
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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
$$ c=\min_{0\leq t\lt\pi}\frac{5+6\cos(t)+3\cos(2t)}{6(\pi-t)^2}=0.069\dots. $$