22 problems

For each integer $m\geq 0$, define … and … Here an empty sum is understood to be zero. Chen's conjecture. For every integer $m\geq 0$, both $U_m(t)$ and $V_m(t)$ are completely mon…

Raşa's conjecture. The function $\psi_{n,c}$ is logarithmically convex. The paper proves complete monotonicity of $\psi_{n,c}$, which implies logarithmic convexity; thus this conje…

For $\lambda \leq -1$, let $\phi_{\lambda}^{-1}$ denote the strict inverse of the power-divergence copula generator $\phi_{\lambda}$, defined on $t \in [0,\infty)$. Complete monoto…

Nested derivative cones conjecture. For every $n\in\mathbb{N}$, the semialgebraic sets $D_k$ form a descending chain under inclusion,

The reverse-inequality conjecture. For every $\alpha\in(1,2)$,

Gaussian completely monotone conjecture. For all $m\geq1$ and all $t>0$,

Qiu, Ma and Huang's conjecture. The function $F$ is completely monotonic on $(0,1/2)$.

Clark–Ismail's coefficient-positivity conjecture. Then $\gamma(j,k)\ge0$ for all $j>1$ and $k\ge0$. This is a coefficientwise strengthening of the positivity question for $g_j$ and…

Clark–Ismail's positivity conjecture. The functions $f_j(v)$, or equivalently $g_j(v)$, are positive on $(0,\infty)$ for all $j>1$. This conjecture concerns the positivity needed t…

Let $n\in\mathbb{N}$ and define the function $x\mapsto x^n/(1-e^{-x})$ on $(0,\infty)$. Clark and Ismail's positivity conjecture. For all $x\in(0,\infty)$ and $n\in\mathbb{N}$, … T…

Let $R_n(x)$, for $n\geq 0$, be the remainders of the asymptotic expansion of $\log\Gamma(x)$, defined by … For a completely monotonic function $f$ on $(0,+\infty)$, its completely…

Qi–Agarwal conjecture. The function

The Le Roy-type Mittag-Leffler function is associated, when the relevant complete-monotonicity condition holds, with an underlying positive random variable. Logarithmic infinite-di…

Parameter-relation conjecture. There must be a relation between the parameters such that the fractional relaxation function is completely monotone, as occurs when the derivative or…

Let $\mathbb{N}$ denote the positive integers and, for $(n,t)\in\mathbb{N}\times[0,+\infty)$, define … The maximum value … should satisfy $\alpha>0.298$. This problem concerns the…

Let $\mu(c)$ denote the parameter arising in the explicit solution of the fundamental Robin eigenvalue problem for a rectangle or box, as defined in the paper. Complete monotonicit…

Complete monotonicity conjecture. The function $h_1$ is completely monotonic on $(0,\infty)$. This is a stronger regularity assertion about a logarithmic ratio related to the gamma…

Mainardi's bounding conjecture. For every $t>0$ and every $\alpha\in(0,1)$, these two rational functions provide upper and lower bounds for $e_\alpha(t)$.

Yu's conjecture. $D(n,\lambda/n)$ is completely monotonic as a function of $n$.

Let $y>-\frac12$ and let $H_y$ denote the function defined in equation of the source, with parameter $\alpha$, on $x\in(-y-1,\infty)$. A function is logarithmically completely mono…

Gamma-quotient complete monotonicity conjecture. For every $n\in\mathbb{N}$ and $x>\frac1a$,

Let $n\geq 1$ be an integer, let $\psi^{(n)}$ denote the $n$-th derivative of the digamma function, and let a function be completely monotonic on $(0,+\infty)$ when all of its deri…