Finally, we obtain some more general conclusions than the first work of this paper, which reveal the absolute monotonicity of four functions involving the four inequalities mentioned above.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper, we first determine the relationships between the first Wilker's inequality, the second Wilker's inequality, the first Huygens inequality, and the second Huygens inequality for circular functions and for hyperbolic functions, respectively.
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Key Findings & Data:Then, we establish new Wilker-type inequalities and Huygens-type inequalities for two function pairs, $x/\sin^{-1}x$ and $x/\tan ^{-1}x$, $x/\sinh ^{-1}x$ and $x/\tanh ^{-1}x$.
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Conclusion & Impact:Finally, we obtain some more general conclusions than the first work of this paper, which reveal the absolute monotonicity of four functions involving the four inequalities mentioned above.
In this paper, we first determine the relationships between the first Wilker's inequality, the second Wilker's inequality, the first Huygens inequality, and the second Huygens inequality for circular functions and for hyperbolic functions, respectively. Then, we establish new Wilker-type inequalities and Huygens-type inequalities for two function pairs, $x/\sin^{-1}x$ and $x/\tan ^{-1}x$, $x/\sinh ^{-1}x$ and $x/\tanh ^{-1}x$. Finally, we obtain some more general conclusions than the first work of this paper, which reveal the absolute monotonicity of four functions involving the four inequalities mentioned above.
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