Inequalities for 1/(1-cos(x)) and its derivatives

Abstract

We prove that the function $g(x)=1/(1-\cos (x))$ is completely monotonic on $(0,\pi ]$ and absolutely monotonic on $[\pi ,2\pi )$, and we determine the best possible bounds ${\lambda }_n$ and ${\mu }_n$ such that the inequalities $$ λ_n \leq g^{(n)}(x)+g^{(n)}(y)-g^{(n)}(x+y) \quad (n \geq 0 \,\,\, \mbox{even}) $$ and $$ μ_n \leq g^{(n)}(x+y)-g^{(n)}(x)-g^{(n)}(y) \quad (n \geq 1 \,\,\, \mbox{odd}) $$ hold for all $x,y\in (0,\pi )$ with $x+y\le \pi$.

8 pages