In using variants of the so-called Bernoulli inequality, new sharp bounds for circular and hyperbolic functions are proved as well as for their products and ratios. We provide some improvements of previous results by using infinite products and power series expansions.

中文翻译：

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使用伯努利不等式类型的圆函数和双曲函数的尖锐不等式

在使用所谓的伯努利不等式的变体时，证明了圆形和双曲函数及其乘积和比率的新锐界。我们通过使用无限乘积和幂级数展开对先前结果进行了一些改进。