In the article, we prove that the function
$r\mapsto \mathcal{E}(r)/S_{9/2-p, p}(1, r')$
is strictly increasing on
$(0, 1)$
for
$p\leq7/4$
and strictly decreasing on
$(0, 1)$
for
$p\in [2, 9/4]$
, where
$r'=\sqrt{1-r^{2}}$
,
$\mathcal{E}(r)=\int_{0}^{\pi/2}\sqrt{1-r^{2}\sin^{2}(t)}\,dt$
is the complete elliptic integral of the second kind, and
$S_{p, q}(a, b)=[q(a^{p}-b^{p})/(p(a^{q}-b^{q}))]^{1/(p-q)}$
is the Stolarsky mean of a and b. As applications, we present several new...
