A concavity property of the complete elliptic integral of the first kind

ABSTRACT

We prove that the function $G_a\left(x\right)=\frac{a-\log \left(\sqrt{1-x}\right)}{\mathcal{K}\left(\sqrt{x}\right)}(a\in \mathbb{R})$ is strictly concave on $(0,1)$ if and only if $a\ge 8/5$. This solves a problem posed by Yang and Tian and complements their result that $1/G_a$ $(a\ge 0)$ is strictly concave on $(0,1)$ if and only if $a=4/3$. Moreover, we apply our concavity theorem to present several functional inequalities involving $\mathcal{K}$. Among others, we prove that if $a\ge 8/5$, then $\frac{2a}{\pi }+1<\frac{a-\log \left(r^{\prime }\right)}{\mathcal{K}\left(r\right)}+\frac{a-\log \left(r\right)}{\mathcal{K}\left(r^{\prime }\right)}\le \frac{2a+\log \left(2\right)}{\mathcal{K}\left(1/\sqrt{2}\right)}$ for all $r\in (0,1),$ where $r^{\prime }=\sqrt{1-r^2}$. Both bounds are sharp and the sign of equality holds if and only if $r=1/\sqrt{2}$.

KEYWORDS:

• Complete elliptic integral of the first kind
• concavity
• hypergeometric function
• inequalities

AMS CLASSIFICATIONS:

• 26D07
• 33C05
• 33E05

Acknowledgements

We thank the referee for helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

ORCID

Kendall C. Richards  http://orcid.org/0000-0002-7424-5129