For $a,b>0$ with $a\neq b$, the Stolarsky means are defined by% \begin{ equation*} S_{ p,q}\left(a,b\right) =\left({ \dfrac{ q(a^{ p}-b^{ p})}{ p(a^{ q}-b^{ q})}}% \right) ^{ 1/(p-q)}\text{ if}pq\left(p-q\right) \neq 0 \end{ equation*}% and $S_{ p,q}\left(a,b\right) $ is defined as its limits at $p=0$ or $q=0$ or $p=q$ if $pq\left(p-q\right) =0$. The complete elliptic integrals of the second kind $E$ is defined on $\left(0,1\right) $ by% \begin{ equation*} E\left(r\right) =\int_{ 0}^{ \pi /2}\sqrt{ 1-r^{ 2}\sin ^{ 2}t}dt. \end{ equation*}% We prove that the functions% \begin{ equation*} F\left(r\right) =\frac{ 1-\left(2/\pi \right) E\left(r\right)}{ % 1-S_{ 11/4,7/4}\left(1,r^{ \prime}\right)}\text{ and}G\left(r\right) =% \frac{ 1-\left(2/\pi \right) E\left(r\right)}{ 1-S_{ 5/2,2}\left(1,r^{ \prime}\right)} \end{ equation*}% are strictly decreasing and increasing on $\left(0,1\right) $, respectively, where $r^{ \prime}=\sqrt{ 1-r^{ 2}}$. These yield some very accurate approximations for the complete elliptic integrals of the second kind, which greatly improve some known results.
Categories: math.CA