Conjectures on convexity and absolute monotonicity of auxiliary elliptic-integral functions

Let $f$, $G$, $h_{11}$ and $h_{12}$ be the functions introduced in the source, let $f_7$ be the function from Lemma 1(7), and define

$$h_{13}(r)=r^{-2}\log\left(\frac{2\mathscr{K}(r)}{\pi}\right),\qquad r\in(0,1).$$

Auxiliary-function conjectures. (1) The function $f$ is convex on $(0,1)$. (2) The functions $h_{11}$ and $h_{12}$ are both strictly increasing and convex on $(0,1)$, with ranges $(79/960,\log(\pi/2))$ and $(517/604800,\infty)$, respectively. (3) The coefficients of the Maclaurin series expansions of $f$, $G$, $f_7$, $h_{11}$, $h_{12}$ and $h_{13}$ are all positive; in particular, these functions are all absolutely monotone on $(0,1)$.

These claims arise from computed series expansions and are intended to provide stronger regularity and positivity properties for the functions used in the elliptic-integral inequalities. The supplied text gives no proof or resolution.