New Cusa-Huygens type inequalities

Ling Zhu; Ling Zhu

doi:10.3934/math.2020341

AIMS Mathematics

2020, Volume 5, Issue 5: 5320-5331. doi: 10.3934/math.2020341

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Research article

New Cusa-Huygens type inequalities

Ling Zhu ^,

Department of Mathematics, Zhejiang Gongshang University, Hangzhou 310018, P. R. China

Received: 24 April 2020 Accepted: 16 June 2020 Published: 22 June 2020
MSC : 26D05, 26D15, 33B10

Using the monotone form of the L'Hôspital rule, we discuss the (absolute) monotonicity of the functions $U\left(x\right) = \frac{1}{x^{4}}-% \frac{1}{x^{5}}\frac{3\sin x}{\cos x+2}$, $G(x) = \frac{1}{x^{2}}\left[\frac{% \ln\sin x-\ln x}{\ln\left(2+\cos x\right) -\ln 3}-1\right]$ and $J(x) = \frac{% 1-(\sin x)/x}{1-(2+\cos x)/3}$ to improve the Cusa-Huygens inequality in several directions on wider ranges. Our results are much better than those existing ones.
- Cusa-Huygens type inequality,
- circular function,
- Bernoulli number
Citation: Ling Zhu. New Cusa-Huygens type inequalities[J]. AIMS Mathematics, 2020, 5(5): 5320-5331. doi: 10.3934/math.2020341

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Abstract

Using the monotone form of the L'Hôspital rule, we discuss the (absolute) monotonicity of the functions $U\left(x\right) = \frac{1}{x^{4}}-% \frac{1}{x^{5}}\frac{3\sin x}{\cos x+2}$, $G(x) = \frac{1}{x^{2}}\left[\frac{% \ln\sin x-\ln x}{\ln\left(2+\cos x\right) -\ln 3}-1\right]$ and $J(x) = \frac{% 1-(\sin x)/x}{1-(2+\cos x)/3}$ to improve the Cusa-Huygens inequality in several directions on wider ranges. Our results are much better than those existing ones.

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