Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters
Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters
In the article, we present the best possible parameters $\lambda=\lambda (p)$ and $\mu=\mu(p)$ on the interval $[0, 1/2]$ such that the double inequality $$\begin{aligned}& G^{p}\bigl[\lambda a+(1-\lambda)b, \lambda b+(1-\lambda)a \bigr]A^{1-p}(a,b) \\& \quad< E(a,b) < G^{p}\bigl[\mu a+(1-\mu)b, \mu b+(1-\mu)a \bigr]A^{1-p}(a,b) \end{aligned}$$ holds for any $p\in[1, \infty)$ and all $a, b>0$ with $a\neq …