In this paper, we present Padé approximations of some functions involving complete elliptic integrals of the first kind $K(x)$ , and motivated by these approximations we also present the following double inequality: $$ \frac{1-x^{2}}{1-x^{2}+\frac{x^{4}}{62}}< \frac{2 e^{\frac{2}{\pi }K(x)-1}}{ (1+\frac{1}{\sqrt{1-x^{2}}} )}< \frac{1-\frac{96}{100}x^{2}}{1-\frac{96}{100}x^{2}+\frac{x^{4}}{64}},\quad x\in ( 0,1 ). $$ Our results have superiority over some new recent results.

中文翻译：

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第一类完整椭圆积分的一些Padé逼近和不等式

在本文中，我们给出了涉及第一类完整椭圆积分的某些函数的Padé逼近$ K（x）$，并基于这些逼近，还提出了以下双重不等式：$$ \ frac {1-x ^ {2 }} {1-x ^ {2} + \ frac {x ^ {4}} {62}} <\ frac {2 e ^ {\ frac {2} {\ pi} K（x）-1}} { （1+ \ frac {1} {\ sqrt {1-x ^ {2}}}）} <\ frac {1- \ frac {96} {100} x ^ {2}} {1- \ frac {96 } {100} x ^ {2} + \ frac {x ^ {4}} {64}}，\ quad x \ in（0,1）。$$我们的结果优于最近的一些新结果。