19 problems

Let $\Psi(a,c,z)$ be the function used in the paper's Laplace-transform formulas, and let CM mean completely monotone on $(0,\infty)$. The complete-monotonicity criterion. For ever…

Let $S_{\alpha}$ be a positive $\alpha$-stable random variable with density $g_{\alpha}$, where $\alpha\in(0,1]$. A density is hyperbolically completely monotone (HCM) if it has th…

Let $F(s)$ be the Laplace transform expression … For $1 \leq \gamma < 2$, the parameters satisfy $s,K>0$, and $F(s)$ is associated with the first-passage-time problem. Complete mon…

Let $r$ denote Mill's ratio, and write $r^{(n)}$ for its $n$th derivative. A function is completely monotone (CM) if all derivatives exist on $(0,\infty)$ and satisfy the alternati…

Let ${\bf B}_{a,b}$ and ${\bf B}'_{a,b}$ denote beta and beta-prime random variables with the indicated positive parameters, and let $\Psi(a,c,z)$ be the function used for the corr…

For integers $m,n$ with $2\leq m\leq n$, let $E_{m,n}(x)=\sum_{1\leq i_1<\dots<i_m\leq n}x_{i_1}\cdots x_{i_m}$ be the $m$th elementary symmetric polynomial, and consider its negat…

Let $p\in \mathbb{R}[x_1,\dots,x_n]$ be a complete hyperbolic polynomial with hyperbolicity cone $C$. For a real $\alpha>n$, let $q_{\alpha}$ be the function defined by … where…

The hexagonal lattice conjecture. The optimizer of this polarization problem is the hexagonal lattice $A_2$. The existence of optimizers is noted under rapid decay assumptions ensu…

Let $f$ satisfy $f(r)=F(r^2)$, where $F$ is a completely monotone function, meaning the Laplace transform of a nonnegative Borel measure. Let $e^\rho$ denote the equidistant altern…

Complete monotonicity conjecture. The stated conditions are sufficient to ensure the existence and complete monotonicity of the solution $\Psi(t)$ for $t\geq 0$. This conjecture co…

Let $S_{\alpha}$ be a positive $\alpha$-stable random variable, let $g_{\alpha}$ denote its density, and define … Let $G_{\alpha}$ denote the density of $S_{\alpha}^{-\beta}$. A fu…

Logarithmic complete monotonicity conjecture. The function $q$ is logarithmically completely monotonic on $(0,\infty)$; that is,

Zero-count conjecture. The number of zeros of

Let $a_1,\ldots,a_p$ and $b_1,\ldots,b_p$ be real parameters, and define … Assume $a_i\ge0$ for $i=1,\ldots,p$ and $\sum_{i=1}^{p}(b_i-a_i)>0$. Zero-count conjecture. The number of…

Let $R$ be the function defined in the source by equation (R(x)), and let $G_a$ be the parameterized function defined earlier in the paper. Monotonicity and complete-monotonicity c…

For $x>0$, let $H(x)$ denote the continuous extension of $H(n)=n^{2}(R_n-\gamma)$, where $R_n=\sum_{k=1}^{n}1/k-\ln(n+1/2)$ and $\gamma$ is Euler's constant. Define the second func…

Clark–Ismail conjecture. The function $\Phi_m^{(m)}(x)$ should be completely monotone on $(0,\infty)$ for every integer $m\geq 1$.

For $\alpha\in\mathbb{R}$, let $D_{\mathbb{R}}$ be the set of parameters such that $\det_{\alpha}(A)\geq0$ for every real symmetric positive semidefinite matrix $A$, and let…

Positivity conjecture. Each polynomial $(-1)^{n-1}p_n(w)$ has all positive coefficients. Consequently, $dW(x)/dx$ is a completely monotonic function.