Mathematics Problem Archive

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...

Suppose that $|z_k | = 1$ $(1 \leq k < \infty)$. Put \[A_l=\limsup_{m\to\infty}\big|\sum^m_{k=1}z_k^l\big|.\] It is easy to see that there is a sequen...

Suppose that $A$ and $B$ are disjoint linked Jordan curves in $\mathbb{R}^3$ which lie at a distance $1$ from each other. Show that the length of $A$ ...

The expression $u(z, \zeta)$ of Problem 6.57 is closely related to the Schwarzian derivative $\{f(z), z\}$, e.g. it is invariant under compositions wi...

Let $E = \{|z|< 1\}$, let $0$, $a > 0$, $b = |b|e^{i\beta}(-\pi < \beta\leq\pi)$ be distinct points of $E$; and let $\mathcal{K}$ be the family of con...

Let $K$ be a compact set of positive measure in $\mathbb{C}$. Does there necessarily exist a non-constant analytic function in $\mathbb{C}\setminus K$...

Is there a homeomorphism of the open unit ball in $\mathbb{R}^3$ onto $\mathbb{R}^3$, whose coordinate functions are harmonic? In other words, do ther...

For a domain $D$ in $\mathbb{C}$, define \[\rho(x,y) = \sup\{|f(x)-f(y)| : x,y\in D; f \text{ analytic in }D; |f'|\leq1\text{ in }D\}.\] If $D$ is con...

Suppose that $f(z)$ is continuous in a domain $D$ and that either [(i)] ; $\int_{|\zeta-z|=r}f(\zeta)\,d\zeta=0$ for all $z\in D$ and $0<r\leq r(z)$, ...

Let $f(z)$ be continuous on $\mathbb{D}$, and let $\alpha$ be a fixed number with $0 < \alpha \leq 1$. If, for each $z$ in $\{|z|< 1\}$, \[\int_{|\zet...

Let $u(z)$ be a real bounded continuous function on $U = \{|z|< 1\}$, and suppose that to each $z \in U$ there corresponds a real number $r(z)$ with $...

Suppose that \[a_1>0,\hspace{1cm}0\leq a_n\leq n,\hspace{1cm} n\geq1,\hspace{1cm} b_n=\sum^n_{\nu=1}a_\nu,\hspace{1cm} c_n=\sum^n_{\nu=1}b_\nu.\] Then...

Let $\mu(t)$ be a continuous monotonic increasing function of $t$ for $t\in[0,1]$ and let $\omega_1(h,\mu)$, $\omega_2(h,\mu)$ denote its modulus of c...

Let $P(\theta)=\sum^N_{n=1}e^{i\lambda_n\theta}$ be a finite Dirichlet series with exponents $\gamma_m\neq\gamma_n$ for $m\neq n$. What can be said ab...

Let $\beta_j\in\mathbb{R}^+$ and $\zeta_j\in\mathbb{C}$, and for suitable small $z$ define $$ f(z)=\prod^n_{j=1}(1-\zeta_jz)^{\beta_j}=1+a_1z+a_2z^2+\...

According to Fefferman's theorem (no citation), a real function $u$ on the unit circle which has bounded mean oscillation, can be decomposed as $u=b_1...

This problem is equivalent to Problem 7.9 due to Gehring and Reich about best bounds for area distribution under quasiconformal mapping. Let $E$ denot...

Let $T$ denote the class of all rational functions $g$ of the form \[g(z)=\sum^n_{j=1}\frac{\lambda_j}{(z-z_j)^2},\] where the constants $\lambda_j$ s...

The Hankel matrices of a function $f$ having a Taylor expansion \[f(z)=a_0+a_1z+\ldots\] are defined by \[H^{(n)}_p=(a_{ij});\hspace{1cm}a_{ij}=a_{n+1...