25 problems

Let $E(r)$ be the complete elliptic integral of the second kind, let $r^{\prime}=\sqrt{1-r^{2}}$, and let $A_{p}(a,b)=\left((a^{p}+b^{p})/2\right)^{1/p}$ for $p\neq 0$, with…

Let $x\in(0,1)$ and let $\mathcal{K}$ denote the complete elliptic integral of the first kind. Anderson–Vamanamurthy–Vuorinen's conjecture. The inequality … holds for all…

Ausserlechner's conjecture. The Hall-plate integral satisfies

Let $f_{\overline{G}_8}(z)$ be the irreducible internally completed graphical function in four dimensions associated with the graph $\overline{G}_8$, with all edges of weight $1$.…

Yang and Tian's conjecture. The function $h_p$ is log-concave on $(0,1)$ if and only if $p\geq 7/32$.

Let the incomplete arithmetic-geometric mean construction produce a rectangular lattice of values of $\arcsin(z)/AGM$, with lattice points obtained by choices of signs for the recu…

Let $r_n$ and $s_n$ denote the roots occurring in the AGM recursion, let $a_\infty$ and $u_\infty$ be its limiting quantities, and let $\sin_p^{-1}$ denote the principal value of t…

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 $W$ be a crystallographic finite reflection group. Let $\delta_W$ be its discriminant, let $k(W)$ be the exponent specifying the root under consideration, and write $\lambda_W$…

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…

Conjectured evaluation. The value of this integral is

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…

All-genus resummation conjecture. By looking at the small $y$ expansion, one conjectures that

Vuorinen–Madras conjecture. For all $a,b>0$ with $a\ne b$,

Gosper–Bickford conjecture. The surface area satisfies

For $0<r<1$, let $\mathcal{K}(r)$ denote the complete elliptic integral and let $\operatorname{arth} r$ denote the inverse hyperbolic tangent. Refined elliptic-integral exponent co…

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…

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 $n\geq 2$ be an integer, and consider the two triples of transcendental equations determining $n$-mode equilibrium ring shapes with points of contact: the system consisting of…

Seventh susceptibility-combination conjecture. The appropriate linear combination giving the minimum-order ODE for $tild\chi^{(7)}$ is

Let $\tilde{\chi}^{(n)}$ be the $n$-particle contribution, and let the lower-index contributions $\tilde{\chi}^{(n-2k)}$ be removed with coefficients $\alpha_{n-2k}$, where…

Let $L_E$ be the linear differential operator corresponding to the complete elliptic integral $E$, and let $L_n$ denote a linear differential operator associated with the $n$-parti…