7 problems

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…

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…

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…