Convexity and concavity of a class of functions related to the elliptic
functions
Convexity and concavity of a class of functions related to the elliptic
functions
We investigate the convexity property on $(0,1)$ of the function $$f_a(x)=\frac{{\cal K}{(\sqrt x)}}{a-(1/2)\log(1-x)}.$$ We show that $f_a$ is strictly convex on $(0,1)$ if and only if $a\geq a_c$ and $1/f_a$ is strictly convex on $(0,1)$ if and only if $a\leq\log 4$, where $a_c$ is some critical value. The second main …