Mathematics Problem Archive

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...

(Subadditivity problem for analytic capacity) Prove or disprove the existence of a constant $M$ such that \[\gamma(K_1\cup K_2)\leq M\{\gamma(K_1)+\ga...

Let $D(z, r)=\{w:|w-z|\leq r\}$. Given a sequence $\{D(z_j, r)\}^N_{j=1}$ of disjoint closed balls all contained in $|z|\leq\frac{1}{2}$, put \[\Omega...

Let $\{z_\nu\}$ be a sequence of distinct points in the unit disc $\mathbb{D}$ such that $\sum(1-|z_\nu|)<\infty$. Let $B$ be the Blaschke product cor...

Define the Harnack function $H_{z_0}(z)$ for a Green domain $D$ relative to $z_0\in D$ to be the supremum of all positive harmonic functions $h$ on $D...

A bounded simply-connected domain $D$ is said to be conformally rigid if there is some $\varepsilon>0$ such that if $f$ is a conformal self-map of $D$...

Let $U$ be an open set in the plane and $\lambda^U_a$ be the harmonic measure at the point $a$ with respect to $U$. Then \O{}ksendal showed that $\lam...

Let $D$ be the unit disc cut along $p$ radial slits from the outer boundary, all of the same given length. Let $u$ be the harmonic measure of $\{|z|=1...

(A `Universal' Phr\'agmen-Lindel\"of Theorem) Let $D$ be an arbitrary unbounded plane domain. Suppose that $f(z)$ is analytic on $D$ and continuous on...

Let $K\subset\mathbb{C}$ be compact and let $x_0\in K$ be a non-peak point for $R(K)$, the uniform limits on $K$ of rational functions with poles outs...

A domain $D\subseteq \mathbb{R}^n$ is said to be linearly accessible, if each point in the complement of $D$ can be joined to $\infty$ by a ray which ...

Let $x=(x_1, x_2, \ldots, x_n)$ be a point in Euclidean $n$-space, $n\geq3$, with $|x|=\Big(\sum^n_{i=1}x^2_i\Big)^{1/2}$. A function $u$ on $\mathbb{...

Let $U$ be a connected open set in $\mathbb{R}^n$. Brelot and Choquet showed that the set of points on the boundary of $U$ which are accessible from t...

Let $D$ be a bounded strictly pseudo-convex domain in $\mathbb{C}^n, n>1$, with smooth boundary. Denote by $A^\infty(D)$ the class of functions analyt...

Let $K\subset\mathbb{C}^n$ be a compact set and let $P_0(K)$ be the set of all polynomials on $K$. The $P$-hull of $K$, the polynomial convex hull of ...

[(i)] ; (One-dimensional version.) Let $E$ be a compact set in $\mathbb{R}$; and, for each $x \in E$, let $\delta_x > 0$ be given. Let $I_x = (x-\delt...

Let $\phi_t(z)= e^{tz}-1$, $\phi^1_t=\phi_t$ and $\phi^{n+1}_t=\phi_t\circ\phi^n_t$ for $n\geq1$; it follows that \[\phi^1_t(-1)=e^{-t}-1 = -t+\frac{t...

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...

Let $\Gamma$ be a Fuchsian group in $\mathbb{D}$, and let $i(z) \equiv z$. Is it true that \[\sum_{\gamma\in\Gamma}|\gamma'(0)|\geq\prod_{\gamma\in\Ga...

Let $\mathcal{W}$ be a hyperbolic Riemann surface, $G_\omega$ the Green's function with pole $\omega\in\mathcal{W}$, and $\Gamma=\{[\gamma_n]\}$ the f...

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...

Let $V$ be the zero set of some analytic function in a strictly pseudo-convex domain $\Omega$ in $\mathbb{C}^2$. If $V$ has finite area inside $\Omega...

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...