Mathematics Problem Archive

Let $f$ be continuous in $\overline{\mathbb{D}}$ and analytic in $\mathbb{D}$. Let \[\omega(f,\delta)=\sup|f(z)-f(w)|,\hspace{1cm}\text{for }|z-w|\leq...

A $G_\delta$ set is a subset of a topological space that is a countable intersection of open sets. Let $E$ be a $G_\delta$ set of measure zero on $|z|...

Let $\mathcal{B}$ be the space of Bloch functions, that is the space of functions analytic in $\mathbb{D}$ with \[\|f\|_\mathcal{B}=|f(0)|+\sup_\mathb...

It was shown by Becker that \[\big\{f:\|f\|_\mathcal{B}<1\big\}\subset \mathcal{B}_Q.\] Is the radius 1 best possible? Is it true that for $f \in \mat...

Suppose that $f_n\in\mathcal{B}_S$. What does $\|f_n-f\|_\mathcal{B}\to0$ as $n\to\infty$ mean geometrically for the functions $g_n$ related to $f_n$ ...

Let $L$ be a regular triangular lattice in the plane. Let $f(z)$ map $\mathbb{D}$ onto the universal covering surface over the complement of $L$. Is i...

It was proved by Hall that every Bloch function has (possibly infinite) angular limits on an uncountably dense subset of $|z | = 1$. Do there always e...

Let $F$ be any discontinuous group of M\"obius transformations of $\mathbb{D}$. Does there always exist a meromorphic function automorphic with respec...

Let $(n_k)$ be a sequence of positive integers such that $$ n_{k+1}>\lambda n_k,\text{ where }\lambda>1, $$ and suppose that $$ f(z)=\sum^\infty_{k=0}...

Suppose that $f(z)$ is a function as in ([source label: B5.13]) and define \[\mu=\limsup_{r\to1}\frac{\log\log M(r,f)}{-\log(1-r)},\] where $M(r,f)$ i...

Tao-Shing Shah has shown that if $g(z)\prec f(z)$ in $\mathbb{D}$, $g'(0)/f'(0)$ is real, and $f$ in $S$ then $$ |g(z)|\leq|f(z)|\text{ for }|z|\leq\f...

Goluzin has shown that, if $g(z)\prec f(z)$ in $\mathbb{D}$, then \[M_2(r,g')\leq M_2(r,f'),\hspace{1cm}0\leq r\leq\frac{1}{2}.\] Here, for $p > 0$, \...

Suppose that $f(z) = \sum^\infty_0 a_nz^n$ and that $F(z)$ is analytic in $\mathbb{D}$, with $f\prec F$. What non-trivial conditions on $F$ imply that...

Suppose that \[f(z)=\sum^\infty_{n=0}a_nz^n\] is analytic in $\mathbb{D}$, with \[\sum^\infty_{n=0}|a_n|=1,\hspace{1cm} |f(z)|\geq\delta>0\text{ in }\...

Determine the Laurent coefficient bodies for analytic functions taking values of modulus at most unity in a given annulus \[A_r = \{z : r < |z| < 1\}....

Suppose that $0 < \alpha < 1$ and \[\frac{(1+xz)^\alpha}{1-z}=\sum^\infty_{n=0}A_n(x)z^n,\hspace{1cm} A_0(x)=1,\hspace{1cm} |x|=1.\] Is it true that \...

If $A$ is any analytic subset of the Riemann sphere it was shown by Kierst that $A$ is (exactly) the set of asymptotic values of a function meromorphi...

A non-constant function $f$ analytic in $\mathbb{D}$, is said to be in the MacLane class $\mathcal{A}$ if the set of points of the unit circle $\mathb...

Let \[f(z)=\sum^\infty_{n=0}a_nz^{\lambda_n}\] be analytic in $\mathbb{D}$ and have Hadamard gaps, i.e. $\lambda_{n+1}/\lambda_n\geq q>1$. Need $f$ ha...

Let \[f(z)=\sum^\infty_{k=1}a_{n_k}z^{n_k}\] be analytic in $\mathbb{D}$ and have Hadamard gaps. Characterise sets $S$ in $\mathbb{D}$ with the proper...

What kind of gaps can the Taylor expansion of a non-constant automorphic function have? For example, can it have Hadamard gaps? (Presumably the sharp ...

A function $f$ analytic in $\mathbb{D}$, is said to be annular if there exists a sequence $\{J_n\}$ of Jordan curves in $\mathbb{D}$ such that [(a)] ;...

It can be shown that there exists a Blaschke product $B(z)$ with \mbox{$B(0) = 0$} such that \[\frac{1+B(z)}{1-B(z)}\] is a Bloch function. Give an ex...

Let F be a Fuchsian group in $\mathbb{D}$, and let $B$ be a set on $\mathbb{T}$ such that $B\cap \gamma(B)=\emptyset$ for $\gamma\in\Gamma,\gamma\neq ...

The ratio $R$ of two Blaschke products $B(z,a_n), B(z,b_n)$ is of bounded characteristic, but need not be a normal meromorphic function. That is, if $...

Suppose that \[f(z)=\sum^\infty_{n=0}a_nz^n\] is convergent in $\mathbb{D}$, that $|z_0| = 1$, and that neither of the series \[\sum^\infty_{n=0}(\tex...

Let the function $h(z)$ be analytic in $\mathbb{D}$, $G_h = \{h(\mathbb{D})\}$, $K_h$ a compact subset of $G_h$. The function $h(z)$ will be said to h...

By a first-order property we shall mean a ring-theoretic property in the first-order predicate calculus. For functions $f$ analytic in $\mathbb{D}$, a...

Suppose that $f$ is analytic in $\mathbb{D}$. Plessner's theorem asserts that at almost all points $e^{i\theta}$ on the unit circle, either $f$ has an...

Suppose that $f$ is univalent and zero-free in $\mathbb{D}$. It has been shown by Baernstein that, for each $p\in(0,\frac{1}{2})$, $f$ admits a factor...

(Subordination and extreme point problem) Let $g(z)=\sum^\infty_{n=0}B_nz^n$ be analytic in $\mathbb{D}$. Denote by $S_g$ the family of functions $f(z...

(Hadamard convolutions) Suppose that $\alpha\geq1, \beta\geq1$ and that $\phi$ is analytic in $\mathbb{D}$, and satisfies \[\phi(z)\ast\frac{(1+xz)^\a...

Let $w(z)$ be analytic in $\mathbb{D}$ with $w(0)=0$. If \mbox{$|w(z)+zw'(z)|<1$}, for $|z|<1$, then a simple application of Schwarz's lemma shows tha...

Let $u$ be a continuous real-valued function on the unit circle $\mathbb{T}$. Give a necessary and sufficient condition on $u$ such that $u$ is the re...

One of the many equivalent norms on BMOA on $\mathbb{D}$ is defined by \[\|f\|_h=\inf_q\sup_{z\in\mathbb{D}}|f(z)+\overline{q(z)}|,\] the infimum bein...

Let $f$ be analytic in $\mathbb{D}$ with $$ |f(z)|=O((1-|z|)^{-k}),\hspace{1cm}k\geq0. $$ Then $f$ induces a distribution on $C^\infty(T)$, as follows...

Does there exist a non-constant function $f$ in the disc algebra such that $f(e^{i\theta})\in f(\mathbb{D})$ for almost all $\theta$? Caution: the Rud...

Let $B$ be an infinite Blaschke product in $\mathbb{D}$. Does there exist a positive $\delta$, depending on $B$, such that, for every $w$, $|w|<\delta...

Let the function $f$ in $\mathbb{D}$ be given by \[f(z)=\sum^\infty_{k=0}a_kz^{n_k},\hspace{1cm}\frac{n_{k+1}}{n_k}\geq\lambda>1,\hspace{1cm} k\geq0,\...

Let the function $f$ where \[f(z)=1+\sum^\infty_{n=1}a_nz^n,\hspace{1cm}|z|\leq1,\] be a Bloch function with positive real part in $\mathbb{D}$. Deter...

Let the function $f$ where \[f(z)=1+\sum^\infty_{n=1}a_nz^n,\hspace{1cm}|z|\leq1,\] be a Bloch function with positive real part in $\mathbb{D}$ and su...

Barth and Clunie have constructed a bounded analytic function in $\mathbb{D}$ with a level set component of infinite length; this component is highly ...

Suppose that \[f(z) = \sum^\infty_{k=1}a_kz^{n_k}, \hspace{1cm}n_{k+1}/n_k\geq q>1,\] is an analytic function in $\mathbb{D}$ with Hadamard gaps, such...

Let the function $f$ have the power series $f(z)=\sum^\infty_{n=0}a_nz^n$ of radius of convergence $1$; let $E$ be the singular set on $\mathbb{T}$, a...

Let $0<\alpha<1$ and let $R_\alpha$ denote the set of all Riesz potentials $p(x)$ of finite positive Borel measures $\mu$ on $\mathbb{R}$: \[p(x)=\int...

Characterise those non-negative measurable functions $f$ on the unit circle that are dominated almost everywhere by moduli of the boundary values of a...

Does there exist a bounded analytic function in $\mathbb{D}$ such that the image of every radius has infinite length? See, for example, Anderson , and...

Let $\gamma$ be a non-tangential arc that lies in $\mathbb{D}$ except for one endpoint at $z = 1$, and define \[\gamma_\theta=e^{i\theta\gamma},\hspac...

In general, the radial behaviour of the derivative of a bounded analytic function in $\mathbb{D}$ can be pretty arbitrary; in fact, even under much st...

Let $H^1$ denote Hausdorff one-dimensional measure on $\mathbb{C}$, and $\mathbb{T}$; let $g:\mathbb{T}\to[-\infty,\infty]$ denote an arbitrary Borel ...