Convexity and concavity of a class of functions related to the elliptic functions

We investigate the convexity property on $(0,1)$ of the function $$f_a(x)=\frac{\mathcal{K}(\sqrt{x})}{a-(1\mathrm{/}2)\log (1-x)}\mathrm{.}$$ We show that $f_a$ is strictly convex on $(0,1)$ if and only if $a\ge a_c$ and $1\mathrm{/}{f}_a$ is strictly convex on $(0,1)$ if and only if $a\le \log 4$, where $a_c$ is some critical value. The second main result of the paper is to study the log-convexity and log-concavity of the function $$h_p(x)=(1-x{)}^p\mathcal{K}(\sqrt{x})\mathrm{.}$$ We prove that $h_p$ is strictly log-concave on $(0,1)$ if and only if $p\ge 7\mathrm{/}32$ and strictly log-convex if and only if $p\le 0$. This solves some problems posed by Yang and Tian and complete their result and a result of Alzer and Richards that $f_a$ is strictly concave on $(0,1)$ if and only if $a=4\mathrm{/}3$ and $1\mathrm{/}{f}_a$ is strictly concave on $(0,1)$ if and only if $a\ge 8\mathrm{/}5$. As applications of the convexity and concavity, we establish among other inequalities, that for $a\ge a_c$ and all $r\in (0,1)$ $$\frac{2\pi \sqrt{\pi }}{(2a+\log 2)\mathrm{\Gamma }(3\mathrm{/}4{)}^2}\le \frac{\mathcal{K}(\sqrt{r})}{a-\frac{1}{2}\log (r)}+\frac{\mathcal{K}(\sqrt{1-r})}{a-\frac{1}{2}\log (1-r)}<1+\frac{\pi }{2a},$$ and for $p\ge 3(2+\sqrt{2})\mathrm{/}8$ and all $r\in (0,1)$ $$\sqrt{(r-r^2{)}^p\mathcal{K}(\sqrt{1-r})\mathcal{K}(\sqrt{r})}<\frac{\pi \sqrt{\pi }}{2^{p+1}\mathrm{\Gamma }(3\mathrm{/}4{)}^2}<\frac{r^p\mathcal{K}(\sqrt{1-r})+(1-r{)}^p\mathcal{K}(\sqrt{r})}{2}\mathrm{.}$$