New Cusa-Huygens type inequalities

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.

Keywords

• cusa-huygens type inequality
• circular function
• bernoulli number