24 problems

Let $\{x_n\}_{n\geq 0}$ be a sequence of nonnegative numbers, and define $z_n=\sum_{k=0}^n A_{n,k}x_k$ using the Eulerian triangle $\{A_{n,k}\}_{0\leq k\leq n}$. A sequence…

Let $f=e^h$ be a smooth, radial, log-convex density on $\mathbb{R}^n$, and define the weighted volume and perimeter of a set $A\subset\mathbb{R}^n$ by … A set is an isoperimetric r…

Amdeberhan–Moll–Vignat conjecture. Both sequences $\{a_n\}_{n\ge 1}$ and $\{b_n\}_{n\ge 1}$ are log-convex. The conjecture concerns recursively defined integer sequences arising al…

Eulerian transformation conjecture. The Eulerian transformation preserves log-convexity. The source presents this as a related problem after discussing q-log-convexity of Eulerian…

Narayana transformation conjecture. The Narayana transformation preserves log-convexity. The source presents this as a problem following known log-convexity results for specializat…

Best-constant conjecture. The best constant $c_2$ in condition (c-2) for the Bell numbers $\{b_k(n)\}_{n=0}^{\infty}$ of any order $k\geq 2$ is $c_2=2$. The surrounding discussion…

For a fixed integer $m\geq 2$, let $H_n^{[m]}(1,1)$ denote the $(1,1)$-Hoggatt sums. The $(1,1)$-Hoggatt-sum infinite log-convexity conjecture. For every fixed $m\geq 2$, the seque…

Let $\{B_n\}_{n\geq 1}$ be the sequence of Baxter numbers. The Baxter-number infinite log-convexity conjecture. The sequence … is infinitely log-convex. The paper proves eventual…

Log-convex density conjecture. Balls centered at the origin provide isoperimetric regions of every prescribed weighted volume. Chambers subsequently proved the assertion for densit…

Let $\Omega$ be a finite set, let $S\colon\Omega\times\Omega\to\mathbb R_+$ be a symmetric matrix with nonnegative entries, and let $u\colon\Omega\to\mathbb R_+$ be a nonnegative u…

Triple-bubble type conjecture. Only three types of perimeter minimizers occur for some densities and four types occur for others.

Smoothness-removal conjecture. The smoothness assumption of Theorem 1 can be omitted.

The Apéry numbers are the sequences … A sequence is infinitely log-convex when every sequence obtained by iterating the log-convexity operator remains log-convex. Chen–Xia's conjec…

Let $H_n(3)$ denote the number of $n\times n$ $(0,1)$-matrices with every row and column sum equal to $3$. Yang's conjecture. The sequence $\{H_n(3)\}_{n\geq 2}$ is log-convex, nam…

Let $\{P_n\}_{n\geq 0}$ be the Catalan–Larcombe–French sequence. A sequence is infinitely log-convex if every sequence obtained by repeatedly applying the operator…

Let $\mathcal{R}$ act on a sequence $\{S_n\}_{n\geq 0}$ by … Let $\{V_n\}_{n\geq 1}$ be the Fennessey-Larcombe-French sequence. Alternating $V_n$ ratio conjecture. For all integer…

Alternating ratio-sequence conjecture. For all integer $k\geq 1$, the sequence $\mathcal{R}^k(\{P_n\}_{n\geq 0})$, except for the first $k$ terms at the beginning, is log-concave i…

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…

For $a,b,\alpha,\beta>0$ and $x\tsim(0,1)$, write the relevant generalized Turán expression as a power series whose coefficients are denoted by $\psi_k$. Coefficient-wise first-par…

A sequence transformation maps a sequence $\{z_n\}_n$ to a sequence $\{w_n\}_n$. For the Jacobi-Stirling numbers of the second kind $\operatorname{JS}(n,k;z)$ and the Jacobi-Stirli…

Let A=a_i_{0\leq i\leq\infty} be a sequence, and define … A sequence is $k$-log-convex if $\mathcal{L}^j(A)$ is log-convex for $j=0,1,\ldots,k-1$, and infinitely log-convex if…

Let $n$, $k$, $a$, and $b$ be integers such that $n\geq k\geq 0$ and $a>b>0$. The sequence … is defined by these binomial coefficients. Su and Wang's log-concavity-to-log-convexity…

Liu–Wang's conjecture. For every fixed $n$ and $r$, there is an integer $k'=k'(n,r)$ such that $\alpha(n,r,k)\geq 0$ for $k\leq k'$ and $\alpha(n,r,k)\leq 0$ for $k>k'$. Consequent…

Let $N(n,k)=\frac{1}{n}\binom{n}{k}\binom{n}{k+1}$ be the Narayana numbers. For a sequence $(a_k)_{k\geq 0}$ of positive real numbers, define its Narayana transform by … A sequence…