Mathematics Problem Archive

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 ...

[(a)] ; Let $f$ be a non-constant analytic function in $\mathbb{D}$, $m$ be a positive integer, and define $\psi = (f)^mf'$. Then it is shown by Sons ...

The Bieberbach conjecture Is it true that $|a_n|\leq n$ for $f$ in $S$ with equality only for $f(z)\equiv f_\theta(z)$? The result is known to be true...

Define $A_n=\sup_{f\in S}|a_n|$. It is shown by Hayman that \[\frac{A_n}{n}\to K_0,\hspace{1cm}\text{ as }n\to\infty.\] Is it true that $K_0=1$? The b...

If $f(z)$ in $S$ it is shown by Bombieri , that there exist constants $c_n$ such that for $f(z)$ in $S$ \[|\text{Re}\,(n-a_n)|\leq c_n\text{Re}\,(2-a_...

If it proves too difficult to obtain sharp bounds for all of the coefficients in Problem 6.4, we ask for the orders of magnitude. An area principle sh...

What are the orders of magnitude of the $c_n$ in Problem 6.4? Springer obtained the estimate \[|c_n|\leq\frac{2^n}{n}\] and also showed that, given $\...

If $f(z)$ in $S$ and is bounded, i.e. satisfies $|f(z)|<M$ for $z\in\mathbb{D}$, we again ask for the order of magnitude of the coefficients $a_n$. Si...

We write \[I_\lambda(r,f)=\Big\{\frac{1}{2\pi}\int^{2\pi}_0|f(re^{i\theta}|^\lambda\,d\theta\Big\}^{1/\lambda}.\] What are the exact bounds for $I_\la...

(Schoenberg's conjecture) If $f(z)=\sum^\infty_{n=1}a_nz^n$ and $g(z)=\sum^\infty_{n=1}b_nz^n$ are convex, and $f$, $g$ belong to $S$, is it true that...

If $F(z)$, $G(z)$ are convex functions in $\Sigma$, it is known that for $0<\lambda<1$, \[H(z)=\lambda F(z)+(1-\lambda)G(z)\in\Sigma,\] see Pommerenke...

If $f(z)$, $g(z)$ are convex functions in $S$, is it true that for $0<\lambda<1$, $\lambda f+(1-\lambda)g$ is star-like and univalent? A function $w=f...

If $f(z)=z+\sum^\infty_{k=2}a_{n_k}z^{n_k}\in S$, and \[\liminf_{k\to\infty}\frac{n_{k+1}}{n_k}>1,\] then Pommerenke has proved that $$ a_n=o\Big(\fra...

Suppose that $f(z)$ in $S$, and that positive integers $k, m, n,$ are given. It is known that there exist complex numbers $c_0, c_1,\ldots,c_m,$ depen...

If $f(z)$ in $S$, set \[A^{(k)}_n= \begin{vmatrix} a_n,&a_{n+1},&\ldots,&a_{n+k-1} \hdotsfor{4} a_{n+k-1},&a_{n+k},&\ldots,&a_{n+2k-2} \end{vmatrix}\]...

If $f(z)$ in $S$, write \[f_\alpha(z)=\int^z_0f'(\zeta)^\alpha\, d\zeta.\] For what values of $\alpha$, is it true that $f_\alpha(z)\in S$? The result...

Let $S^*$ be the class of all star-like functions $f(z)$ in $S$. Marx conjectured that for each fixed $z_0$, $|z_0|<1$, the set of all numbers $f'(z_0...

If $f(z)=z+\sum^\infty_{n=2}a_nz^n$ in $S$, then \[ A=\pi\sum^\infty_{n=1}n|a_n|^2\] is the area of the image domain. What is the minimum value of $A$...

If $F(z)=z+\sum^\infty_{n=1}b_nz^{-n}$ in $\Sigma$, then \[A(F)=\pi-\pi\sum^\infty_{n=1}n|b_n|^2\] is the area of the set of values not assumed by $F(...