Sharp generalized Seiffert mean bounds for the Toader mean of order 4

In the article, we present the best possible parameters $$\alpha ,\beta $$ such that the double inequality $$\begin{aligned} S_{\alpha }(a,b)0$$ 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.