On approximating the quasi-arithmetic mean
On approximating the quasi-arithmetic mean
In this article, we prove that the double inequalities $$\begin{aligned} &\alpha_{1} \biggl[\frac{7C(a,b)}{16}+\frac{9H(a,b)}{16} \biggr]+(1- \alpha_{1}) \biggl[\frac{3A(a,b)}{4}+\frac{G(a, b)}{4} \biggr]\\ &\quad< E(a,b) \\ &\quad< \beta_{1} \biggl[\frac{7C(a,b)}{16}+\frac{9H(a,b)}{16} \biggr]+(1- \beta_{1}) \biggl[\frac{3A(a,b)}{4}+\frac{G(a, b)}{4} \biggr], \\ &\biggl[\frac{7C(a,b)}{16}+\frac{9H(a,b)}{16} \biggr]^{\alpha _{2}} \biggl[ \frac{3A(a,b)}{4}+\frac{G(a, b)}{4} \biggr]^{1-\alpha_{2}}\\ &\quad< E(a,b) \\ &\quad< \biggl[\frac{7C(a,b)}{16}+\frac{9H(a,b)}{16} \biggr]^{\beta _{2}} \biggl[ \frac{3A(a,b)}{4}+\frac{G(a, b)}{4} \biggr]^{1-\beta_{2}} \end{aligned}$$ hold for …