A functional bound for Young's cosine polynomial. II.

We prove that $$ \sum_{k=1}^{2\lfloor\frac{n}{2}\rfloor+1}\frac{(-1)^{k-1}}{k}+\sum_{k=1}^{n}\frac{\cos k\theta}{k}\geqslant\frac{1}{4}\left(1+\cos\theta\right)^2\qquad(n=1,2,3,\ldots;\,\theta\in\left(0,\pi\right)), $$ where equality holds if and only if $n=2$ and $\theta=\pi-\cos^{-1}\frac{1}{3}$. This refines inequalities due to Alzer et al. and Fong et al.