Mathematics Problem Archive

Given a domain of holomorphy $\Omega\subseteq\mathbb{C}^n$, $n\geq2$, what conditions are required on $\Omega$ to admit an analytic function $p:\Omega...

Let $N$ denote the class of complex-valued $L^\infty$-functions $v$ on the unit disc $U$ such that $\int_U v\phi\,dx\,dy=0$ whenever $\phi$ is analyti...

Let $D_1$, $D_2$ be domains in $\{|z|< R\}$, and let $\lambda_1(z)\,|dz|$ and $\lambda_2(z)\,|dz|$ be their hyperbolic metrics. What is the least numb...

In their famous Acta paper , Hardy and Littlewood introduced the celebrated Hardy-Littlewood maximal function in connection with complex function theo...

Prove or disprove the following statements about analytic capacity $\gamma$. [(a)] ; If $E\subseteq\mathbb{C}$ is compact and $\phi$ is a $C^1$-diffeo...

Prove or disprove the following statement: If $K$ is a compact subset of $\mathbb{C}$ with continuous analytic capacity $\alpha(K)= 0$, then \[\alpha(...

We will say that $g(x)$ is a rearrangement of $f(x)$ if \[m\{x:g(x) < y\} = m\{x:f(x) < y\}\hspace{1cm}\text{ for all } y\in\mathbb{R},\] where $m$ is...

Does there exist a sequence $\{z_n\}^\infty_1$ of distinct complex numbers such that \[\sum\frac{1}{|z_n|}<\infty\hspace{1cm}\text{and}\hspace{1cm}\su...

Let $f$ be analytic on a domain $G$ in $\mathbb{C}$. We will say that a point $z_0\in G$ is a MacLane point of $f$ if there exists some neighbourhood ...

Let $f$ be an inner function, with $f(0) = 0$; then $f$ induces an ergodic (Lebesgue-) measure-preserving map of the circle onto itself. What is the e...

Let $I$ denote the class of all inner functions. Then $I$, as a subset of $H^\infty$, enjoys some of the properties that the collection of unimodular ...

Let $E\subset \mathbb{C}$, and the function $F:E\times \mathbb{D}\to\mathbb{C}$ satisfy the following conditions: [(a)] ; $F$ is injective on $E$, for...

Let $G$ be a finitely-generated Fuchsian group of the first kind. [(a)] ; If $F:\Pi \to \Pi$ is a $G$-invariant quasi-symmetric function, is $F$ total...

Let $L = \{L\}^\infty_0$ be a non-negative increasing sequence such that \[\sum^\infty_{k=0}L_kr^k<\infty\hspace{1cm}(0<r<1).\] If $f(z)$ is analytic ...

Given functions $f_1,\ldots,f_N\in H^\infty$, let $I = I(f_1,\ldots,f_N)$ be the ideal of $H^\infty$ generated by $f_1,\ldots,f_N$ and let $J = J(f_1,...

For a bounded plane domain $D$, denote by $N(D)$ the class of functions analytic on $D$ of bounded characteristic (i.e., all quotients of functions in...

Let $D$ be a simply-connected domain in the complex plane $\mathbb{C}$ and $A(D)$ the ring of functions $f: D\to\mathbb{C}$ analytic in $D$. Bers (no ...

Let $G$ be a domain in $\mathbb{C}$, and $H(G)$ the ring of functions analytic on $G$. It is known that, for two domains $G_1$ and $G_2$, $H(G_1)$ is ...

Let $A^p$, $p > 0$ be the Bergmann space of functions $f(z)$ analytic in $|z|< 1$ such that \[\|f\|_p=\Huge({\int\int}_{|z|<1}|f(re^{i\theta})|^p\,r\,...

We use the notation of Problem 8.6. Let $\{z_n\}^\infty_1$ be a sequence of points in $\mathbb{D}$ such that the kernel functions $k_n(z) = (1-z_nz)^{...

Suppose that $F$ is a relatively-closed subset of $\mathbb{D}$, and let \[\|f\|_F=\sup\{|f(z)|;z\in F\}\] for functions $f$ in the Bergman space $A^2$...

Determine \[\|\Lambda\|=\sup\Big|{\int\int}_{|z|<1}f(z)\phi(z)\,d\sigma_z\Big|\] over those $f\in A^1$ with $\|f\|_1\leq1$, where \[\phi(z)=\text{sgn}...

If $f$ and $1/f$ belong to the Bergman space $A^2$, does it follow that $\mathcal{P}f$ is dense in $A^2$? Here $\mathcal{P}f$ denotes the set of all p...

Let $g$ be a function in the space $D$ of functions analytic in $|z|< 1$ with finite Dirichlet integral, i.e., \[{\int\int}_{|z|<1}|h'(z)|^2\,dx\,dy<\...

Let $A$ denote the set of functions continuous on $|z|= 1$ which extend continuously to analytic functions on $|z|< 1$ (the disc algebra). Let $\tilde...

Does there exist a singular measure in Zygmund's class $A^*$ all of whose Fourier-Stieltjes coefficients are non-negative? (F. Holland)...

Which functions in $L^\infty$ on the unit circle generate positive Hankel operators? (F. Holland)...

Characterise the Hankel operators on the Hardy space $H^2$ on the circle that are of trace class. (F. Holland)...

Miles and Rudin have shown that in $\mathbb{C}^n$ a function analytic in the unit polydisc and in the Hardy class $H^1$ may not be expressible as the ...

In the ring of bounded analytic functions on the unit ball or the polydisc in $n$ variables, is the intersection of two finitely-generated ideals agai...

For which simply-connected domains $D$ (with $0 \in D$) is it true that there is a constant $K = K(D)$ such that $$ {\int\int}_D|f|^2\,dx\,dy\leq K{\i...

Let $\mu\geq1$ be a singular measure on the boundary of the unit disc $\mathbb{D}$; and let $S_\mu$ be the corresponding singular inner function \[S_\...

Let $f_1,\ldots,f_n$ and $g$ be $H^\infty$ functions on $\mathbb{D}$. If $|g(z)| \leq |f_1(z)| + \ldots + |f_n(z)|$, are there necessarily functions $...

Let $w\geq0$ be an integrable function on the circle or on the line. Then $w$ is said to belong to the class $A_2$ if \[\sup_l\Big(\frac{1}{|l|}\int_l...

Do there exist inner functions in any strictly pseudo-convex domain of $\mathbb{C}^n$, $n\geq2$? Alexandrov (no citation) and independently L{\o}w (no...

In Pompeiu's formula, $f(z)$ and $\int_\Gamma f(\zeta)/(\zeta-z)\,d\zeta$ make sense for merely continuous functions, but \[\int\int_\Omega\frac{\part...

Let the sequence $\{M_k\}^\infty_0$ of positive numbers be such that \[M_0=1\hspace{1cm}\text{ and }\hspace{1cm}\frac{M_{k+j}}{M_kM_j}\geq\begin{pmatr...

Let $\psi:S^1\to S^1$ be a direction-reversing homeomorphism, and let $A_\psi$ denote the set of functions $f:S^1\to\mathbb{C}$ such that both $f$ and...

Let $\psi:S^1\to S^1$ be a homeomorphism. When is it true that $\text{Re }A = \text{Re }A \circ \psi$? That is, when is each function $f$ in the real ...

A sequence $\{z_n\}^\infty_1$ in $|z|< 1$ is interpolating for bounded analytic (harmonic) functions if, for each bounded sequence $\{\alpha_n\}^\inft...

Let $f(z)\in H^\infty$, and let $\{z_n\}$ be a Blaschke sequence \[\sum^\infty_{n=1}(\-|z_n|)<\infty.\] [(a)] ; Does there always exist a Blaschke pro...

[(a)] ; Suppose that $f, f_1, f_2,\ldots,f_n\in H^\infty$, and \[|f|\leq|f_1|+|f_2|+\ldots+|f_n|.\] Do there necessarily exist $h_1,h_2,\ldots,h_n\in ...

For each pair of $f,g\in H^\infty$, does there necessarily exist another pair of functions $a,b\in H^\infty$ such that $$ af+gb\neq0,\hspace{1cm}|z|<1...

Let $K_1, K_2, K_3$ be disjoint closed sets in the extended complex plane, and $C_1, C_2, C_3$ constants. Let $\rho_n(f)$ be the best rational approxi...

Let $D$ be an open subset of the extended complex plane with non-empty boundary $\partial D$, and let $F$ be a relatively-closed subset of $D$. Let $f...

Let us call a closed set $E$ in $\mathbb{C}$ a weak Arakelian set if, corresponding to each function $g(z)$ continuous on $E$ and analytic in the inte...

Let $\gamma$ be a Jordan arc in $\mathbb{C}^n$, $n\geq2$ such that the projections $\gamma_j$ on the complex coordinate planes $j=1,\ldots,n$ have are...

Does the condition $\sum 1/p_n<\infty$ for positive integers $p_n$ guarantee that the sequences of powers $\{z^{p_n}\}$ fails to span $C(\gamma)$ for ...

Let $F$ be a closed subest of $\mathbb{R}^n$, $n\geq2$. Call $F$ a set of harmonic approximation if every function continuous on $F$ and harmonic in t...

Let $D$ be a planar domain. A sequence $\{z_j\}$ of points in $D$ is said to be an interpolating sequence if whenever $\{\alpha_j\}\in\ell^\infty$ the...