373 problems

Finite-difference stability conjecture. For any stable polynomial $p$, the polynomials

Zero-strip-decreasing conjecture. The operator $\Delta_{\theta,h}$ defined on $\mathbb{C}[z]$ is zero strip decreasing.

For every normalized univalent harmonic mapping $f=h+\overline{g}$ of the unit disk $\mathbb{D}$, with expansions $h(z)=z+\sum_{n=2}^{\infty}a_nz^n$ and…

For every entire function $f:\mathbb{C}\to\mathbb{C}$, is every subset of $\mathbb{C}$ definable, with parameters, in the structure $(\mathbb{C};+,\cdot,0,1,f)$ either countable or…

Let $2\le n<N$, and let $F:\mathbb{B}^n\to\mathbb{B}^N$ be a proper holomorphic map admitting a $C^{N-n+1}$-smooth extension to the boundary. If the CR Gauss map of the boundary re…

For every integer $n\ge 0$ and every nonzero polynomial $p$ of degree $n$ whose zeros all lie on the unit circle $\mathbb{T}$, define $Q_n(z)=1+z^n$. For all $0\le s\le t\le\infty$…

The supplied source identifies Borcea's $2$-variance conjecture as the $p=2$ case of a conjecture concerning the distance between the critical points of a polynomial and those of i…

For $\nu>0$, let $J_\nu$ denote the Bessel function of the first kind, let $j_{\nu,n}$ be its $n$th positive zero, and define the normalized Turán expression by…

Let $\mathbb{D}^2=\{z\in\mathbb{C}^2:|z_1|<1,\ |z_2|<1\}$, let $\alpha>0$, and let $\mathcal{D}_\alpha(\mathbb{D}^2)$ be the corresponding Dirichlet-type space. For every polynomia…

Let $A^2_\alpha$ be the weighted Bergman space on $\mathbb{D}$, with $\alpha>-1$, and let $M\subseteq A^2_\alpha$ be invariant under the multiplication operator $S f(z)=z f(z)$. De…

Let $f\colon\mathbb{C}\to\mathbb{P}^n$ be a transcendental linearly non-degenerate holomorphic curve with finite lower order, and let $N_1(r,f)$ denote the counting function of the…

For every separable Hilbert space $\mathcal E$, every $B\in H^\infty_1(L(\mathcal E))$, and every $\alpha>-1$, the associated sub-Bergman spaces satisfy…

For every $M>1$, let $f:\mathbb{D}\to\mathbb{C}$ be harmonic and satisfy $|f(z)|<M$ for all $z\in\mathbb{D}$, together with the normalization $f(0)=0$, $f_z(0)=1$, and…

For every integer $n\ge 1$, every real number $\rho>1$, and all complex numbers $z_1,\ldots,z_n$ satisfying $|z_j|\le \rho$ for $1\le j\le n$, prove that…

For every entire function $F$ of order $\rho$ with $0<\rho<\tfrac12$, the deficiency of $0$ as a value of its logarithmic derivative vanishes:…

Let $p_n(z)=\prod_{j=1}^n(z-z_j)$, where $z_1,\ldots,z_n$ are independent with common law $\mu_0$ on $\mathbb{C}$, and let $p_{n,t}$ denote the holomorphic heat evolution of $p_n$…

Let $n\geq 2$, let $\mathbb{B}^n$ be the complex unit ball with its canonical Kähler–Einstein metric $g_{\mathbb{B}}$, and let $\Omega_1,\ldots,\Omega_m$ be irreducible bounded sym…

For weighted Bergman spaces $A_{\alpha}^{p}$, determine a necessary-and-sufficient characterization of the zero-free functions $f\in A_{\alpha}^{p}$ whose reciprocal $1/f$ also bel…

Let $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$. If $f=h+\overline{g}:\mathbb{D}\to\mathbb{C}$ is a normalized sense-preserving harmonic mapping with bounded convex image, where…

Let $\Omega\subset\mathbb{R}^{d+1}$ be the Lorentz future cone, let $T=\mathbb{R}^{d+1}+i\Omega$, and let $M\geq 1$. For every connected unipotent subgroup…

Let $R(t,w)\in\mathbb{C}(t,w)$ be rational in $t$ and $w$, and assume that $R$ is not a polynomial in $w$ of degree at most $2$. If a function $f$ meromorphic on $\mathbb{C}$ satis…

Given a closed complex submanifold $C$ and a closed totally real submanifold $R$ of $\mathbb{C}^n$, determine necessary and sufficient conditions for $C\cup R$ to admit a Stein nei…

For every integer $N\ge 1$, every collection of distinct points $\mathbf a_1,\ldots,\mathbf a_N\in\mathbb R^3$, and every collection of nonzero charges $q_1,\ldots,q_N$ having the…

For every integer $n\geq 2$, every point $x\in\mathbf P^n$, and every $\lambda\in[0,1]$, does there exist a function $u\in DMA(\mathbf P^n,\omega_{\mathrm{FS}})$ such that…

For every $n\ge 2$ and every bounded pseudoconvex domain $\Omega\subset\mathbb{C}^n$ with smooth boundary, does the existence of a nonconstant analytic disc in the boundary, כלומר,…