In this paper, we present new inequalities about hyperbolic functions with much better approximation effect. It firstly provides two-sided bounds of $$(\sinh (x)/x)^p$$ for the case $$p \in (0,1]$$ , and lower bound for the case $$p \ge \frac{7}{5}$$ as well. It also provides inequalities about mixed hyperbolic functions consisting of $$\tanh (x)$$ and $$\sinh (x)$$ . Numerical examples show that the new inequalities can achieve much better approximation effect than those of prevailing methods.

中文翻译：

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基于重新参数化的双曲函数的新不等式

在本文中，我们提出了具有更好近似效果的双曲函数的新不等式。它首先为 $$p \in (0,1]$$ 的情况提供 $$(\sinh (x)/x)^p$$ 的两侧边界，并为 $$p \ge 的情况提供下界\frac{7}{5}$$ 也是如此。它还提供了由 $$\tanh (x)$$ 和 $$\sinh (x)$$ 组成的混合双曲函数的不等式。数值例子表明新的不等式可以达到比主流方法更好的逼近效果。