Optimal bounds for Toader mean in terms of general means

Abstract

AbstractIn this paper, we present the best possible parameters $\alpha (r)$α(r), $\beta (r)$β(r) such that the double inequality $$\begin{aligned} {}[\alpha (r)M^{r}(a,b)+(1-\alpha (r))N^{r}(a,b)] ^{1/r} < &\operatorname{TD}\bigl[M(a,b),N(a,b)\bigr] \\ < &\bigl[\beta (r)M^{r}(a,b)+\bigl(1-\beta (r)\bigr)N^{r}(a,b) \bigr]^{1/r}, \end{aligned}$$ [α(r)Mr(a,b)+(1−α(r))Nr(a,b)]1/r<TD[M(a,b),N(a,b)]<[β(r)Mr(a,b)+(1−β(r))Nr(a,b)]1/r, holds for all $r\leq 1$r≤1 and $a,b>0$a,b>0 with $a\neq b$a≠b, where $$ \operatorname{TD}(a,b):= \int ^{\pi /2}_{0}\sqrt{a^{2}\cos ^{2}\theta +b^{2}\sin ^{2} \theta }\,d\theta $$TD(a,b):=∫0π/2a2cos2θ+b2sin2θdθ is the Toader mean, and M, N are means. As applications, we attain the optimal bounds for the Toader mean in terms of arithmetic, contraharmonic, centroidal and quadratic means, and then we provide some new bounds for the complete elliptic integral of the second kind.