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

TL;DR

This paper investigates the complete and absolute monotonicity properties of the function 1/(1-cos(x)) on specific intervals and establishes optimal bounds for inequalities involving its derivatives.

Contribution

It proves the monotonicity properties of 1/(1-cos(x)) and determines the best bounds for inequalities involving its derivatives.

Findings

g(x) is completely monotonic on (0,π]

g(x) is absolutely monotonic on [π,2π)

Optimal bounds λ_n and μ_n are identified for derivative inequalities

Abstract

We prove that the function $g(x)= 1 / \bigl( 1 - \cos(x) \bigr)$ 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 $$ \lambda_n \leq g^{(n)}(x)+g^{(n)}(y)-g^{(n)}(x+y) \quad (n \geq 0 \,\,\, \mbox{even}) $$ and $$ \mu_n \leq g^{(n)}(x+y)-g^{(n)}(x)-g^{(n)}(y) \quad (n \geq 1 \,\,\, \mbox{odd}) $$ hold for all $x,y\in (0,\pi)$ with $x+y\leq \pi$.