118 problems

Eremenko's conjecture. Each connected component of $I(f)$ is unbounded.

Brück's conjecture. If

Newman's conjecture. All zeros of $\Xi_\aleph(\lambda)$ are real for $\aleph\in\mathbb{R}$. This is the conjecture associated with the de Bruijn–Newman constant, concerning the tra…

Bank–Laine conjecture. If $\rho(A)\notin\mathbb{N}$, then

A transcendental entire function is a transcendental holomorphic map $f:\mathbb{C}\to\mathbb{C}$; its order is the usual growth order of an entire function, and a Fatou component i…

For a non-monomial entire function $f$, let $M(f,r)=\max_{|z|=r}|f(z)|$ and let … Writing … Marchenko's conjecture concerns the possibility that $v_f(r)$ tends to infinity, meaning…

Let $f$ be a real entire function, meaning an entire function real-valued on the real axis, and suppose that both $f$ and $f”$ have only real zeros. Let $LP$ denote the Laguerre–Pó…

Denjoy's conjecture. The number of finite asymptotic values of $f$ is at most $2k$.

Let … where $q$ is a complex number satisfying $0<|q|\leq 1$. Sokal's stronger conjecture. The function $\mathcal F(z;q)$ can have only simple zeros with distinct absolute values.…

Jordan-domain conjecture. Every bounded immediate attracting or parabolic basin of a transcendental entire function is a Jordan domain.

Let $f$ be an entire function of finite order, and let $\{\delta_n\}$ be the sequence of its deficiencies, where $\delta(a,f)$ denotes the Nevanlinna deficiency at…

Infinite $B_0$ conjecture. If $f$ is a transcendental entire function and $F(f)\not=\emptyset$, then

Uniform boundedness conjecture. If $f(z)\in\Delta$ and $F(f)\not=\emptyset$, then $F(f)$ is uniformly bounded.

Extension conjecture. If $a$ is locally omitted by $f$, then $f^{-1}(D)$ is disconnected. Moreover, if $f$ has a completely invariant domain $D$, then every locally omitted value b…

For a non-negative integer $p$, let $V_{2p}$ be the class of entire functions of the form … where $a\leq 0$ and $w$ is a real entire function of genus at most $2p+1$ with all zeros…

Let $f$ be a real entire function, meaning that $f$ maps the real axis into itself. Suppose that the order of $f$ is less than $2$ and that $f$ has only finitely many non-real zero…

Differentiation is better than midpoint conjecture. The gaps between consecutive zeros of $f'$ lie between the infimum and supremum of the corresponding two-step midpoint-process g…

Differentiation conjecture. There exist sequences $A_n$, $B_n$, $C_n$, and $D_n$ with $D_n/C_n$ bounded, such that

Krein class bounded-type conjecture. The function $f$ is of bounded type in $bb\setminus E$ if and only if

Let $f$ and $g$ be two distinct permutable entire functions, and let $Q$ be a non-constant polynomial. Suppose that … where $a\in\mathbb{C}\setminus\{0\}$ and $b\in\mathbb{C}$. Wan…

Let $v$ be a vector field satisfying the assumptions of the Liouville conjecture for stationary Navier–Stokes fields. In addition, suppose that $v$ is the restriction to…

$(Q_n)$ conjecture. $(Q_n)$ is true for all nonnegative integers $n$.

Extremal-constant problem. In the previous problem, $c=q_\infty$.

The remainder characterization conjecture. The sequence ${\mathbf A}$ is a $\mathrm{TP}_\infty$-preserver if and only if, for every $l\in\mathbb N\cup\{0\}$, $R_l[\mathbf A]$ is an…

Weighted Paley–Wiener conjecture. The following equivalences hold: