Mathematics Problem Archive

It is known that any quasi-conformal homeomorphism of $B^n = \{x : x \in\mathbb{R}^n,|x|<1\}$ onto a Jordan domain $D$ in $\mathbb{R}^n$ can be extend...

Let $\Gamma$ be a closed Jordan curve in the extended plane, and suppose that $\infty\in\Gamma$. Let $f_1, f_2$ map the upper and lower half-planes, r...

For $n\geq2$, let \[\mathbb{R}^n_+=\big\{(x_1,\ldots,x_n)\in\mathbb{R}^n:x_n>0\big\}\] and \[\mathbb{R}^n_-=\big\{(x_1,\ldots,x_n)\in\mathbb{R}^n:x_n<...

Let $E\subset[0,1]$ be a compact set on the positive real axis in $\mathbb{R}^2$; and let $E$ have positive conformal $2$-capacity, that is $M\big(\De...

Let $(P, Q)$ denote a pair of polynomials with the following property: $$ \text{the map }f\mapsto P(D)(Qf)\text{ carries }\mathcal{E}\text{ bijectivel...

Let $P$ be a polynomial of degree $m$ in which the coefficient of $z^m_1$ is non-zero, and let $Q(z) = z^m_1$. [(a)] ; Does the pair $(P, Q)$ have the...

Is it true that, for any polynomial $P$ with complex coefficients, the mapping $f\mapsto P^*(D)(Pf)$, where $P^*(z) = \overline{P(\overline{z})}$, is ...

Given a countable number of convergent series with positive decreasing terms, one can find such a series converging more slowly than any of these. Wit...

For $n\geq 2$ and $\alpha > 0$, let \[\widehat{T^\alpha_Rf}(\xi)=\Big(1-\frac{|\xi|^2}{R^2}\Big)^\alpha_+\hat{f}(\xi)\] where $R > 0$ and $f\in\mathca...

A continuous mapping $f:B^n\to\mathbb{R}^n$, where $B^n = \{x : x\in\mathbb{R}^n,|x|<1\}$, and $n\geq2$, is said to be proper if $f^{-1}(K)$ is a comp...

For a Hadamard gap sequence $\{n_k\}^\infty_{k=1}$, $n_{k+1}/n_k\geq q>1$, is it true that the measure in $[0,2\pi[$ of the set of those points $x$ fo...

Given an associative algebra $A$, with identity $1$ and countable basis, then for a finite-dimensional subspace $V$ spanned by the vectors $\{e_j\}^k_...

Using arguments due to Ahlfors (see, for example ) any M\"obius transformation in $\mathbb{R}^n$ can be written in the form $(ax + b)(cx+d)^{-1}$, whe...

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 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)...

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...

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...

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...