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New Refinements of Cusa-Huygens Inequality

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Abstract

In the paper, we refine and extend Cusa-Huygens inequality by simple functions. In particular, we determine sharp bounds for \(\sin x /x\) of the form \((2+\cos x)/3 -(2/3-2/\pi )\Upsilon (x)\), where \(\Upsilon (x)>0\) for \(x\in (0, \pi /2)\), \(\Upsilon (0)=0\) and \(\Upsilon (\pi /2)=1\), such that \(\sin x/x\) and the proposed bounds coincide at \(x=0\) and \(x=\pi /2\). The hierarchy of the obtained bounds is discussed, along with a graphical study. Also, alternative proofs of the main result are given.

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Acknowledgements

We would like to thank the three referees for the careful review on the manuscript. This research was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, Projects 451-03-68/2020/14/200156 (second author), ON 174032 and III 44006 (third author).

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Authors and Affiliations

  1. Department of Mathematics, K. K. M. College Parbhani (M.S.), 431505, Manwath, India

    Yogesh J. Bagul

  2. Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia

    Bojan Banjac

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

    Christophe Chesneau

  4. Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovi’ca 6, 21125, Novi Sad, Serbia

    Marko Kostić

  5. School of Electrical Engineering, University of Belgrade, Belgrade, Serbia

    Branko Malešević

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  1. Yogesh J. Bagul
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  2. Bojan Banjac
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  3. Christophe Chesneau
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  4. Marko Kostić
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  5. Branko Malešević
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Correspondence to Yogesh J. Bagul.

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Bagul, Y.J., Banjac, B., Chesneau, C. et al. New Refinements of Cusa-Huygens Inequality. Results Math 76, 107 (2021). https://doi.org/10.1007/s00025-021-01392-8

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