Geodesic
The Cusa-Huygens inequality revisited
Novi Sad Journal of Mathematics, Tome 50 (2020) no. 2.

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Let $c,\ \gamma \in {\mathbb R},$ $\gamma \geq 1,$ $c\geq 1$ and $T\in (0,\pi/\gamma]$ if $c=1,$ resp. $T\in (0,\pi/2\gamma]$ if $c>1.$ In this paper, we find the necessary and sufficient conditions on $a,\ b \in {\mathbb R}$ such that the inequalities \begin{align*} \frac{\sin x}{x}>a+b\cos^{c}(\gamma x),\quad x\in (0,T) \end{align*} and \begin{align*} \frac{\sin x}{x}+b\cos^{c}(\gamma x),\quad x\in (0,T) \end{align*} 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.
Publié le :
DOI : 10.30755/NSJOM.10667
Classification : 26D05, 26D07, 26D20, 33B30
Keywords: Cusa-Huygens inequality, sinc function, series expansion, l'Hospital's rule of monotonicity 
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     author = {Yogesh J. Bagul and Christophe Chesneau and Marko Kosti\'c},
     title = {The {Cusa-Huygens} inequality revisited},
     journal = {Novi Sad Journal of Mathematics},
     pages = {149 - 159},
     publisher = {mathdoc},
     volume = {50},
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     year = {2020},
     doi = {10.30755/NSJOM.10667},
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Yogesh J. Bagul; Christophe Chesneau; Marko Kostić. The Cusa-Huygens inequality revisited. Novi Sad Journal of Mathematics, Tome 50 (2020) no. 2. doi : 10.30755/NSJOM.10667. http://geodesic.mathdoc.fr/articles/10.30755/NSJOM.10667/

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