New Wilker-type and Huygens-type inequalities

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.