An Optimal Double Inequality between Seiffert and Geometric Means

Abstract

For $\alpha ,\beta \in (\mathrm{0,1}/2)$ we prove that the double inequality $G(\alpha a+(1-\alpha )b,\alpha b+(1-\alpha \left)a\right)<P(a,b)<G(\beta a+(1-\beta )b,\beta b+(1-\beta \left)a\right)$ holds for all $a,b>0$ with $a\ne b$ if and only if $\alpha \le (1-\sqrt{1-4/{\pi }^2})/2$ and $\beta \ge (3-\sqrt{3})/6$. Here, $G(a,b)$ and $P(a,b)$ denote the geometric and Seiffert means of two positive numbers a and b, respectively.