In the article, we present the best possible parameters \(\alpha ,\beta \) such that the double inequality

$$\begin{aligned} S_{\alpha }(a,b)<T_{4}(a,b)<S_{\beta }(a,b) \end{aligned}$$

holds for \(a, b>0\) with \(a\ne b\), and provide new bounds for the complete elliptic integral of the second kind, where \(S_{p}(a,b)\) and \(T_{4}(a,b)\) are the generalized Seiffert mean and Toader mean of order 4, respectively.

中文翻译：

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阶4的Toader均值的尖锐广义Seiffert均值界

在本文中，我们提出了最佳的参数\（\ alpha，\ beta \），以使双重不等式

$$ \ begin {aligned} S _ {\ alpha}（a，b）<T_ {4}（a，b）<S _ {\ beta}（a，b）\ end {aligned} $$

保持\（a，b> 0 \）与\（a \ ne b \），并为第二种完整椭圆积分提供新边界，其中\（S_ {p}（a，b）\）和\（T_ {4}（a，b）\）分别是4阶的广义Seiffert均值和Toader均值。