Optimal inequalities for bounding Toader mean by arithmetic and quadratic means
Optimal inequalities for bounding Toader mean by arithmetic and quadratic means
In this paper, we present the best possible parameters $\alpha(r)$ and $\beta(r)$ such that the double inequality $$\begin{aligned} \bigl[\alpha(r)A^{r}(a,b)+ \bigl(1-\alpha(r) \bigr)Q^{r}(a,b) \bigr]^{1/r} < & TD \bigl[A(a,b), Q(a,b) \bigr] \\ < & \bigl[\beta(r)A^{r}(a,b)+ \bigl(1-\beta(r) \bigr)Q^{r}(a,b) \bigr]^{1/r} \end{aligned}$$ holds for all $r\leq 1$ and $a, b>0$ with $a\neq b$ , and we …