Mathematics Problem Archive

Let $\Omega$ be an arbitrary domain in $\mathbb{C}$. Does there necessarily exist a set $E$ in $\partial\Omega$, of full harmonic measure, with the fo...

If $\mathbb{R}^2_+=\{(x,y)\in\mathbb{R}^2:y>0\}$, suppose that $E\subset\mathbb{R}^2_+$, and let $f:\mathbb{R}^2_+\to B^2$ be analytic and conformal w...

Let the function $f(z) = z + a_2z^2 + \ldots$ map $\mathbb{D}$ univalently onto a domain $\Omega$, and let $F: \Omega\to\mathbb{D}$ denote the inverse...

Let $\Omega$ be a simply-connected domain in $\mathbb{C}$ with at least two boundary points, and let the function $\phi$ map $\Omega$ analytically and...

Determine an intrinsic characterisation for the class $\mathcal{H}$ of functions $h$ analytic in $\mathbb{D}$ that admit a decomposition of the form $...

For $-\infty<p<+\infty$ let \[B(p):=\sup\{\beta_f(p):f\text{ conformal map of }\mathbb{D}\text{ into }\mathbb{D}\}\] where \[\beta_f(p)=\limsup_{r\to1...

Goodman conjectured that if $f(z) = \sum^\infty_{n=1}a_nz^n$ is $p$-valent in $\mathbb{D}$, then for each $n> p$, we have \[|a_n|\leq\sum^p_{k=1}\frac...

The coefficients of a $p$-valent function are bounded by some function of its zeros. In particular, let the function \[f(z)=z^q+\sum^\infty_{n=q+1}a_n...

A function $f(z) = z + a_2 z^2 +\ldots$ is said to belong to the class $CV(R_1,R_2)$ if it is univalent and convex in $\mathbb{D}$, and if on $f(\{|z|...

Given two functions $f$, $g$ in the (usual) class $S$, we can form the new functions (arithmetic and geometric mean functions) \[F(z)=\alpha f(z)+\bet...

Let $K$ be a closed set of points in $\mathbb{C}$, and let $F(K)$ denote the family of functions $f$ of the form \[f(z)=\sum^n_{k=1}\frac{A_k}{z-a_k},...

Let $\{v_n\}^\infty_1$ be a sequence of positive integers (which may include $\infty$); the sequence is called a valence sequence if there is a functi...

The function \[k(z)=2\text{Re }\Big(\frac{z+\frac{1}{3}z^3}{(1-z)^3}\Big)=\sum^\infty_{n=1}\frac{1}{3}(2n^2+1)r^n(e^{in\theta}+e^{-in\theta}),\] where...

It is known that, for functions $f$ in $S^0_H$, $\{|w| < \frac{1}{16}\}\subset f(\mathbb{D})$. Prove that the correct value $d$, of the Koebe constant...

What are the convolution multipliers $\phi^*:K_H\to K_H$, where $K_H$ is the subclass of functions $f$ in $S_H$ with convex images $f(\mathbb{D})$? A ...

Let $J$ be a Jordan curve in $\mathbb{C}$ bounding a domain $D$. Suppose that $f:e^{it}\mapsto f(e^{it})$ is a sense-preserving homeomorphism of the u...

Prove that, for $f\in S^0_H$, $$ \big||a_n|-|a_{-n}|\big|\leq n,\hspace{1cm} n=2,3,4,\ldots. $$ (This is a generalisation of the Bieberbach conjecture...

Let $f$ be analytic univalent in $\mathbb{D}$, and consider \[I_\lambda(r,f')=\Big(\frac{1}{2\pi}\int^{2\pi}_0\big|f'(re^{i\theta})\big|^\lambda\,d\th...

Let $f$ be analytic univalent in $\mathbb{D}$, and consider \[I_{-\lambda}(r,f')=\Big(\frac{1}{2\pi}\int^{2\pi}_0\big|f'(re^{i\theta})\big|^{-\lambda}...

Let $\Omega$ be a simply-connected domain in the finite plane whose complement contains $n$ disjoint closed balls with centres on the interval $[0,1]$...

Let $A$ denote the class of functions $f(z) = z + a_2z^2 +\ldots$ analytic in $\mathbb{D}$. For $\delta\geq0$ and $T= \{T_k\}^\infty_2$ a sequence of ...

If $f$ in $A$ and $\delta > 0$, define a $\Sigma_\delta(f)$ neighbourhood of $f$ to be \[\Big\{g:g\in A,\big|(g'(z)-f'(z))-\frac{1}{z}(g(z)-f(z))\big|...

Following the notation of Problem 6.111 and 6.112, it is known that, if $|x|\leq\rho\leq1$ and $\gamma=1/(1+\rho)^2$, then \[N_\gamma\Big(\frac{z}{1-x...

Let $\Gamma$ be a regular curve and $f$ an analytic and conformal function in the open unit disc. Does $f^{-1}(\Gamma)$ necessarily have finite length...

Let $\Gamma$ be a rectifiable curve, and let $E$ be a subset of $\Gamma$ having zero length. If $\Omega$ is any simply-connected domain and $z\in\Omeg...

Let $D$ be a domain in $\mathbb{C}$ containing the origin $0$; for $t>0$, let $\Omega_t$ be the component of $D \cap\{|z|\leq t\}$ containing $0$. In ...

Let $G$ be a domain in $\mathbb{C}$ that contains the origin $0$ and is axially-symmetric with respect to the real axis, that is, if a point $z\in G$ ...

If $A^{(p)}_n=\sup_{f\in S_p}|a_n|$ is it true that \[\frac{A^{(p)}_n}{n^{2p-1}}\to K_p,\hspace{1cm}\text{ as }n\to\infty,\] and if so, what is $K_p$?...

Here our counter-example shows that $|a_n|=o(n^{-\frac{1}{2}})$ is best possible for bounded $f(z)$ in $S(p)$....

If we ask the analogous problems to those of Problem 6.8 for the class $S(p)$, the correct orders of magnitude are again known in many cases, but not ...

The results of Pommerenke were proved in fact for mean $p$-valent functions, and if $f(z)=\sum^\infty_{n=0}a_nz^n$ is mean $p$-valent with $p>\frac{1}...

Here again the main conclusions extend to mean $p$-valent functions. In this case ([source label: 6.6]) holds with $j_k=-\frac{1}{2}+16(p^3/k)^{\frac{...

Let $E$ be the compact plane set of transfinite diameter ($=$capacity) $d(E)=1$ and let \[d_n(E)^{n(n-1)/2}=\max_{w_\nu\in E}\prod_{1\leq\mu<\nu\leq n...

Let $f(z)$ be analytic in a simply-connected domain $D$. It is known that $f(z)$ can be expanded in a series of Faber polynomials \[f(z)=\sum^\infty_{...

Let $z_i$, $1\leq i\leq n$ be a finite sequence of complex numbers such that $|z_i|\leq1$. Set \[S_k=\sum^n_{i=1}z^k_i.\] Can we have $$ \max_{2\leq k...

If $z_1=1$, and the $z_i$ are arbitrary complex numbers for $2\leq i\leq n$, then Atkinson proved that \[\max_{1\leq k\leq n}|S_k|>c\] with $c=\frac{1...

We consider the range of the random function \[F(z)=\sum^\infty_{n=0}\pm a_nz^n\] ($F$ chosen at random in the natural way) defined in $\mathbb{D}$, w...

Is it possible to express each $K$-quasiconformal map in $3$-space as the composition of two quasiconformal maps with maximal dilatation less than $K$...

Suppose that $f$ is a plane $K$-quasiconformal mapping of the unit disc $\mathbb{D}$ onto itself. Show that there exists a finite constant $b = b(K)$ ...

It was proved by Boyarski\u\i\, that the partial derivatives of a plane $K$-quasiconformal mapping are locally $L$-integrable for $2\leq p < 2+c$, whe...

Show that each quasiconformal mapping of $\mathbb{R}^n$ onto $\mathbb{R}^n$ has a quasiconformal extension to $\mathbb{R}^{n+1}$ . This has been estab...

Suppose that $f$ is an $n$-dimensional $K$-quasi-analytic function. Show that the partial derivatives of $f$ are locally $L$-integrable for $n\leq p\l...

One part of Nevanlinna theory is devoted to the following problem. How does the geometric structure of a simply connected Riemann covering surface of ...

(Boundary values of Cauchy integrals). Let $\gamma$ be a $C^1$ curve in the plane and let $f$ be a continuous function on $\gamma$. Put \[F(z)=\int_\g...

Let $D$ be a domain in the extended complex plane. A finite point $z$ on the boundary $\partial D$ of $D$ is called angular (relative to $D$) if there...

Let $\gamma$ be a Jordan arc, $d\mu$ a measure on $\gamma$. Does the Laplace transform $$ f(z) = \int_\gamma e^{z\zeta}\,d\mu(\zeta) $$ always have `a...

Let $f(z)$ be analytic and bounded for $\Re z> 0$. Suppose that $|\alpha|<\frac{1}{2}\pi$ and that $(r_n)$ is a sequence of positive integers with $\s...

Let $\Gamma$ be a Jordan curve and suppose that $z = 0$ lies inside it. Wermer showed that when $\Gamma$ has infinite length, the powers $z^n$, $n\neq...

For what sets $\Omega$ of lattice points $(m_k, n_k)$ do the monomials $x^{m_k}y^{n_k}$ span $L^2$ or $C_0$ on the unit square $0\leq x\leq1$, $0\leq ...

(Two constant theorems for the polydisc) Let $F(z_1, z_2)$ be defined for $|z_1\leq1, |z_2|\leq2$ except when $z_1 = z_2$ and $|z_1|=|z_2|=1$. Suppose...