10 problems

Let $K(x)$ and $E(x)$ denote the complete elliptic integrals of the first and second kinds, respectively, and let $K'(x)$ and $E'(x)$ denote their complementary counterparts. The i…

Let $\mathcal{K}(r)=(\pi/2)F(1/2,1/2;1;r^2)$ be the complete elliptic integral of the first kind. For $r\in(0,1)$, set $r'=\sqrt{1-r^2}$. Anderson–Vamanamurthy–Vuorinen's conjectur…

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…

Monotonicity conjecture for $f$. The function $f$ is strictly increasing from $(0,1)$ onto $(1/320,1/4)$.

Alzer–Qiu's conjecture. The function $G$ is strictly increasing and convex from $(0,1)$ onto $(3/4,1)$.

Let $\mathcal{K}(r)$ denote the complete elliptic integral of the first kind and define … A function is absolutely monotonic on an interval if all of its derivatives are nonnegativ…

Let $\{V_n\}_{n\geq 0}$ be the Fennessey-Larcombe-French sequence, and consider the sequence $\{V_n^2-V_{n-1}V_{n+1}\}_{n\geq 2}$. Infinite log-convexity conjecture. The sequence…

Let $r^{\prime}=\sqrt{1-r^{2}}$. For $p\neq 0$, let $S_{p,q}(a,b)$ denote the Stolarsky mean, and let $p_{0}\approx 1.763135$ be the unique solution on $(0,9/4]$ of … that is,…

Let $\{V_n\}_{n\geq 0}$ be the Fennessey-Larcombe-French sequence, defined by $V_0=1$, $V_1=8$, and … A sequence $\{a_n\}$ is log-concave when $a_n^2\geq a_{n-1}a_{n+1}$ for every…

Let $K(x)$ be the complete elliptic integral of the first kind, and let $K'(x)$ denote its complementary counterpart. The integral is taken over $0\leq x\leq 1$. Cubic elliptic-int…