5 problems

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…

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

Monotonicity and asymptotics conjecture. If $i$ is a positive odd integer, then $g_i(x)$ is strictly decreasing on $(-1,\infty)$; if $i$ is a positive even integer, then $g_i(x)$ i…

Concavity and convexity conjecture. The function $h(x)$ is strictly concave on $(-1,1)$ and strictly convex on $(1,\infty)$. This conjecture refines the proved fact that $h$ is str…

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…