16 problems

Let $p$ and $q$ be integers with $p<q$, let ${\mathbf b}=(b_1,\ldots,b_q)$ satisfy $b_i>0$, and let ${\mathbf a}=(a_1,\ldots,a_p)$ satisfy $a_i\geq b_i$ for $1\leq i\leq p$. The ge…

Farmer–Rhoades conjecture. The displayed inequality holds for every $P\in\mathcal{LP}1$ with its zeros listed in increasing order and every pair of consecutive zeros $\xi<\eta$ of…

The $TP_\infty$-preserver conjecture. The sequence ${\mathbf A}$ is a $TP_\infty$-preserver if and only if, for every $l\in\mathbb{N}\cup\{0\}$, the formal power series

Let $b_1,b_2>0$, let $\gamma>0$, and write $\alpha_n=N_n/D_n$ for the quantities defined in the surrounding discussion. Sufficient-conditions conjecture. (a) The denominator polyno…

$N$-th Hankel relaxation Riemann hypothesis. The function $\Lambda$ is in the $N$-th order Laguerre–Pólya class. These are proposed relaxations of the Riemann hypothesis: membershi…

Let $f$ be an entire function of genus one with $f(0)\ne 0$, and let $\kappa$ denote its aperture. If $f$ belongs to the $N$-th order Laguerre–Pólya class, the established bound ha…

Let $\phi(x)$ belong to the positive shifted Laguerre–Pólya class $\mathcal{SL\text{-}P}^{+}$, and let $L_j$ be the Laguerre operator defined by the coefficient expansion … The Lag…

Let $\phi(x)$ be a real entire function in the shifted Laguerre–Pólya class $\mathcal{SL\text{-}P}$. For each $d\in\mathbb{N}$, suppose there exists $N_3(d)$ such that … for all…

Let $\phi(x)$ be a real entire function of order less than $2$ with a finite number of non-real zeros. The Pólya–Wiman conjecture. There exists an $N$ such that … for all $m\geq N$…

The real-zero conjecture. All but a finite number of zeros of $f$ are real and simple.

Let … be an entire function. The monotonicity conjecture states that if, for some $a_1>1$, membership of $F_{a_1}$ in the Laguerre–Pólya class is equivalent to the existence of … s…

Let ${}_pF_q(\boldsymbol{a};\boldsymbol{b};z)$ be the generalized hypergeometric function, with $p<q$, positive denominator-parameter vector $\boldsymbol{b}>0$, and numerator and d…

Square-root factorial series conjecture. The function $\varphi$ belongs to the Laguerre–Pólya class with nonnegative coefficients, denoted $\mathscr{L}-\mathscr{P}^{+}$.

Fourier-transform conjecture. The function $F$ belongs to the Laguerre–Pólya class:

Classification conjecture. The operator $\phi(D)$ is a complex zero strip decreasing operator if and only if at least one of the following conditions holds: (i) $\beta>0$; (ii)…

Let $S'_r=V_\alpha$, where $\alpha_0=1$, $\alpha_r=-1$, and $\alpha_i=0$ for $i\notin\{0,r\}$. Let $\mathcal{L}P^+$ denote the class used in the paper of entire functions in the La…