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On the Cusa–Huygens inequality

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Sharp bounds of various kinds for the famous unnormalized sinc function defined by \((\sin x)/x\) are useful in mathematics, physics and engineering. In this paper, we reconsider the Cusa–Huygens inequality by solving the following problem: given real numbers \(a,b, c\in {{\mathbb {R}}}\) and \(T\in (0,\pi /2],\) we find the necessary and sufficient conditions such that the inequalities

$$\begin{aligned} \frac{\sin x}{x}>a+b\cos ^{c}x,\quad x\in (0,T) \end{aligned}$$

and

$$\begin{aligned} \frac{\sin x}{x}<a+b\cos ^{c}x,\quad x\in (0,T) \end{aligned}$$

hold true. In the case \( c=1\), the inequalities are extended on \( (0, \pi ). \) We use the elementary methods, only, improving several known results in the existing literature.

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Acknowledgements

The authors thank the two referees for their detailed and constructive comments.

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Authors and Affiliations

  1. Department of Mathematics, K. K. M. College, Dist : Parbhani, Manwath, M.S., 431505, India

    Yogesh J. Bagul

  2. LMNO, University of Caen-Normandie, Caen, France

    Christophe Chesneau

  3. Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21125, Novi Sad, Serbia

    Marko Kostić

Authors
  1. Yogesh J. Bagul
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  2. Christophe Chesneau
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  3. Marko Kostić
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Correspondence to Christophe Chesneau.

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The Marko Kostić was supported in part by Ministry of Science and Technological Development, Republic of Serbia, Grant no. 174024.

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Bagul, Y.J., Chesneau, C. & Kostić, M. On the Cusa–Huygens inequality. RACSAM 115, 29 (2021). https://doi.org/10.1007/s13398-020-00978-1

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  • DOI: https://doi.org/10.1007/s13398-020-00978-1

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