Abstract
In 1953, Turan published the inequality $$\begin{aligned} \sum _{k=1}^n \frac{1\cdot 3 \cdots (2k-1)}{2\cdot 4 \cdots 2k}\cos (kx)>-1 \quad {(n\in \mathbf {N}; \, 0<x<\pi )}. \end{aligned}$$ We prove a refinement of this result and offer inequalities for the corresponding sine polynomial. Moreover, we present an application to the theory of absolutely monotonic functions.
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