32 problems

Krzyż's conjecture. For every $f\in\Omega$ and every $n\geq 1$,

Boundary characterization conjecture. The boundary $\partial\mathcal{L}_\alpha$ coincides with the set

Let $N$ be the Nevanlinna class of holomorphic functions on the unit disk with bounded characteristic, namely those satisfying … where $\log^+x=\max\{0,\log x\}$. Bloch–Nevanlinna…

For $\Lambda\subset\mathbb N\cup\{0\}$ and a function $f$ holomorphic in a domain $D$ containing $0$, define $\langle f,D\rangle\in P_\Lambda$ if $f$ has a power-series expansion ……

Let $M^n$ be a complete noncompact Kähler manifold of complex dimension $n$ with nonnegative holomorphic bisectional curvature, and let $\mathcal{O}_d(M^n)$ denote the space of hol…

Let $F$ be an infinite sequence of $p$-tuples of holomorphic functions $f=(f_1,\ldots,f_p)$ in the unit disc, with each $f_j$ having no zeros and satisfying … Assume the conclusion…

Let $F$ be an infinite sequence of $p$-tuples $f=(f_1,\ldots,f_p)$ of holomorphic functions in the unit disc, with each $f_j$ having no zeros and satisfying … For a subset…

Let ${\mathcal R}=\bigoplus_{d=0}^{\infty}{\mathcal R}^d$ be the graded algebra with its positive-definite inner product, and let $\widehat{{\mathcal R}}$ be its Hilbert-space comp…

Let $K$ be the compact Lie group considered in the paper, let $K_{\mathbb C}$ be its complexification, let $t>0$, and let $C_t$ denote the generalized Segal–Bargmann transform from…

Let $R$ be a simply connected convex region of a discrete Riemann surface, and let the associated discrete exponentials be holomorphic functions on $R$. Exponential-generation conj…

General restriction conjecture. Theorems 3.3 and 3.4 should be partial cases of a general fact concerning restriction operators for holomorphic functions of polynomial growth to ti…

Yau's conjecture. One has

Let $\mathbb{D}$ be the unit disk, and let $\operatorname{Cara}(\mathbb{D};2,3)$ be the set of ×2- and ×3-circular Carathéodory functions; call an element rational extreme when it…

Let $\mathbb{D}$ be the unit disk, and let $\operatorname{Cara}(\mathbb{D};2,3)$ be the set of Carathéodory functions that are ×2- and ×3-circular, where ×$n$-circular means…

Let $\mathbb{D}$ be the unit disk, and let $\operatorname{Cara}(\mathbb{D};2,3)$ be the set of holomorphic functions $\psi:\mathbb{D}\to\mathbb{C}$ with…

Let $\mathcal{O}_d$ denote the space of holomorphic functions on the gradient Kähler Ricci shrinker under consideration whose growth rate is at most $d$. Munteanu–Wang's conjecture…

Let $\cU$ be the parameter domain, let $\cV$ be the target space, and let $L^{\infty}(\cU;\cV)$ denote the space of essentially bounded $\cV$-valued functions on $\cU$. Suppose the…

Let $\Omega$ be a domain, let $B(\Omega)$ be a Banach space of holomorphic functions on $\Omega$, and let $\mathcal{M}(B(\Omega))$ denote its algebra of multipliers. Multiplier rec…

Let $\Omega$ be a domain and let $B(\Omega)$ be the Banach space of holomorphic functions on $\Omega$. Reciprocal conjecture. If $\psi\in B(\Omega)$ and there exists a constant…

Let $\mathbb{B}_n$ denote the unit ball, and let $A^p_\alpha$ be the weighted Bergman space on $\mathbb{B}_n$, where $0<p<\infty$ and $\alpha$ is real. Zhu's reciprocal conjecture.…

Let $H^p$ be the Hardy space on the unit disk, with $p>1$, and let $f(z)=\sum_{n=0}^\infty c_nz^n$ be a nonvanishing function satisfying $\|f\|_p\leq 1$. Hummel–Scheinberg–Zalcman…

Let $\rho$ be a complex density, let $\rho$ define the Schwinger–Dyson equation, and let $\mathcal{T}$ be the space of finite linear combinations … of linear forms $T_\gamma$ assoc…

Let $M$ be a complete noncompact $n$-dimensional Kähler manifold with nonnegative holomorphic bisectional curvature. Let $O_{P}(M)$ be the ring of all holomorphic functions on $M$…

Let $M$ be a complete noncompact $n$-dimensional Kähler manifold with nonnegative holomorphic bisectional curvature. Let $\mathcal{O}_{d}(M^{n})$ denote the family of holomorphic f…

Let $\mathcal{A}$ be a family of complex holomorphic functions. Let $z\in\mathbb{C}^{n}$, let $U$ be an open neighbourhood of $z$, and let $f:U\rightarrow\mathbb{C}$ be holomorphic…