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The Cusa-Huygens inequality revisited. (English) Zbl 1465.26016

Summary: Let \(c\), \(\gamma\in\mathbb{R}\), \(\gamma\geq 1\), \(c\geq 1\) and \(T\in(0,\pi/\gamma\rbrack\) if \(c=1\), resp. \(T\in(0,\pi/2\gamma\rbrack\) if \(c>1\). In this paper, we find the necessary and sufficient conditions on \(a,b\in\mathbb{R}\) such that the inequalities \[ \frac{\sin x}{x}>a+b\cos^c(\gamma x),\quad x\in (0,T) \] and \[ \frac{\sin x}{x}<a+b\cos^c(\gamma x),\quad x \in (0,T) \] hold true. We also determine the best possible constants \(p\) and \(q\) such that \[ \frac{2+\cos(px)}{3}<\frac{\sin x}{x}<\frac{2+\cos(qx)}{3}, \quad x\in (0,\pi/2). \] The proofs of main results contain several auxiliary results which can be of some independent interest.

MSC:

26D05 Inequalities for trigonometric functions and polynomials
26D07 Inequalities involving other types of functions
26D20 Other analytical inequalities
33B30 Higher logarithm functions
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