Mathematics Problem Archive

Let $L_d$ denote the class of those functions $g\in R_d$ such that every critical point of $g$ is preperiodic but not periodic. Show that, if the func...

Let a function $f$ in $R_d$ have the property that \[f(z)=\lambda_\alpha z+O(z^2)\hspace{1cm}\text{ as }z\to0,\] where $\lambda_\alpha=e^{2\pi i\alpha...

Does there exist a number $\lambda$ of modulus one that is not a root of unity, such that the positive orbit of $-\frac{1}{2}$ under $P_ \lambda(z) = ...

Suppose that $\lambda$ is of modulus one and not a root of unity, let $P_ \lambda(z) = \lambda(z + z^2)$ and \[h_\lambda(z)=z+O(z^2)\] is the unique f...

Let the function $f(z)$, $f(z) = \lambda(e^z-1)$ with $|\lambda| = 1$, have a Siegel singular disc $S_\lambda$ that contains zero. [(a)] ; Prove that ...

Does there exist a non-linear entire function $g$ with wandering domain $W$ such that $\bigcup_{n\geq0}g^n(W)$ is bounded in $\mathbb{C}$? It has been...

Let the function $f_0$ in $R_d$ have an invariant Herman singular ring $A_f$ of rotation number $\alpha$, where $\alpha$ satisfies a diophantine condi...

Does there exist a number $\alpha$ in $\mathbb{R}\setminus\mathbb{Q}$ that does not satisfy a diophantine condition, such that every $\mathbb{R}$-anal...

Under the same conditions as in Problem 2.12, is it true that if $\rho\Delta<1$, $f(z)$ cannot have a finite asymptotic value? This is known if $\rho\...

If $u(z)$ is harmonic in the plane, and not a polynomial, does there exist a path $\Gamma_n$ for every positive integer $n$, such that $$ \frac{u(z)}{...

If $u(x)$ is harmonic and not constant in space of $3$ or more dimensions, is it true that there exists a path $\Gamma$ such that $u(x)\to+\infty$ as ...

Suppose that $u(z)$ is subharmonic and $u(z)<0$ in the half-plane $|\theta|<\pi/2$, where $z=re^{i\theta}$. Suppose also that \[A(r)=\inf_{|\theta|<\p...

Consider the class of functions subharmonic in the unit disc $\mathbb{D}$, and satisfying $u(z)\leq0$ there. Suppose also that $A(r,u)\leq-1$, for $r$...

Suppose that $u(z)$ is positive and subharmonic in $\mathbb{D}$, and that there exists a series of arcs $\gamma_n$ tending to the arc $\alpha\leq\thet...

It follows from a result of Wolf , that if \[u(re^{i\theta})\leq f(\theta),\hspace{1cm} 0<r<+\infty,\] where \[\int^{2\pi}_0\log^+f(\theta)\,d\theta<+...

Problem 1.17 can be reformulated for subharmonic functions, if we replace $\log M(r,f)$ by a general subharmonic function $u(z)$. The same positive th...

If $D$ is a convex domain in space of $3$ or more dimensions, can we assert any inequalities for the Green's function $g(P,Q)$ of $D$ which generalise...

Suppose that $u(X)$ is harmonic on the unit ball $|X|<1$, and remains continuous with partial derivatives of all orders on $|X|=1$, where $X$ is a poi...

If $u(x)$ is a homogeneous harmonic polynomial of degree $n$ in $\mathbb{R}^m$, what are the upper and lower bounds of \[-\frac{A(r,u)}{B(r,u)},\] whe...

Consider a domain of infinite connectivity in $\mathbb{R}^3$ whose complement $E$ lies in the plane $P : x_3 =0$. Suppose further than any disc of pos...

Let $u(x)$ be subharmonic in $\mathbb{R}^m$. One can define the quantities $n(r, 0), N(r, 0), T(r)$ as in Nevanlinna theory in the plane, taking the a...

Let there be given an integrable function $F$ on $\mathbb{T}$ and a point $z_0$ in $\mathbb{D}$. The problem is to maximise $u(z_0)$, where $u$ runs t...

Let $D$ be a doubly-connected domain with boundary curves $\alpha$ and $\beta$ and let $z_0, z_1$ be points of $D$. Let $A, B$ be given real numbers. ...

A compact set $E$ in $\mathbb{R}^n$, $n\geq3$ is said to be thin at $P_0$ if $$ \int^1_0\frac{c(P_0,r)}{r^{n-1}}\,dr<\infty, $$ where $c(P_0,r)= \text...

Let $D, D'$ be Lipschitz domains in $\mathbb{R}^n$, $n\geq 3$ with $D' \subset D$ and $\partial D'\cap\partial D$ lying compactly in the interior of a...

It is known that the set $E$ of least capacity $C$ and given volume is a ball. If $E$ displays some measure of asymmetry (for instance, if every ball ...

Let $C_0$ be a tangential path in $\mathbb{D}$ which ends at $z=1$, and let $C_\theta$ be any rotation of $C_0$. Littlewood showed that there exists a...

Suppose that you have a continuous real function $u(x)$ on $\mathbb{R}^n$, and you want to know whether a homeomorphism $\phi:\mathbb{R}^n\to\mathbb{R...

Let $\alpha$ be a continuum in the closure of the unit disc $\mathbb{D}$, and let $\omega(z) = \omega(z; \mathbb{D}; \alpha)$ be the harmonic measure ...

Let $D$ be a domain containing the origin whose `outer boundary' is $\mathbb{T}$ and whose `inner boundary' is a closed set $E$ in $\mathbb{D}$. If ev...

Determine whether or not there exists a function $g(r)$, defined for $r\geq0$, with $g(r)\to0$ as $r\to\infty$, such that the following holds: if $u$ ...

For which positive $p$ does there exist a function $u$, $u\not\equiv0$ harmonic on $\mathbb{R}^3$ and vanishing on the cone $x^2_1+x^2_2=px^2_3$? (H. ...

Is there a harmonic polynomial $P(x_1, x_2, x_3)$, $P\not\equiv 0$ that is divisible by $x^4_1+x^4_2+x^4_3$? (H. S. Shapiro)...

Given $n, n\geq4$, find a continuous function $f$ on $(0,1)$ such that the following statement is true: if $u$ is a subharmonic function in the unit b...

Let $D$ be an unbounded domain in $\mathbb{R}^n$, $n\geq2$. Is there a positive continuous function $\varepsilon(|x|)$ such that, if $u$ is harmonic i...

Determine all domains $\Omega$ in $\mathbb{R}^n$, $n\geq2$, satisfying the identity $\int_\Omega h(x)\,dx = 0$ for every function $h$ harmonic and int...

It is known that the Newtonian potential of a uniform mass distribution spread over an ellipsoid $K$ in $\mathbb{R}^n$, $n\geq2$ is a quadratic functi...

Let $K(z, z')$ denote the kernel of the double layer potential occurring in Fredholm's theory where $z, z' \in \Gamma$, $\Gamma$ being a smooth Jordan...

Let $D$ be an unbounded domain in $\mathbb{R}^n$, $n\geq 2$. Points in $\mathbb{R}^n$ will be denoted by $x = (x_1,x_2,\ldots,x_n)$, and $|x|$ will de...

Let $\Omega$ be an open ball in $\mathbb{R}^n$, $n\geq 2$. It is shown by Armitage that $V\in L^p(\Omega)$ for any positive superharmonic function $V$...

Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$, $n\geq 2$, and suppose that $y\in\partial\Omega$. Denote the open ball of centre $y$ and radi...

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$, $n\geq 2$, with the property that there exists $\alpha$ in $(0,\pi]$ such that for every point $y$...

For $r_1<r_2$, we will call the set $\{x\in\mathbb{R}^n:n\geq3,r_1<\|x\|<r_2\}$ an annulus and its closure a closed annulus. Let $\Omega$ be a non-emp...

Let $\{z_n\}, 1\leq n<\infty$ be an infinite sequence such that $|z_n|=1$. Define \[A_n=\max_{|z|=1}\prod^n_{i=1}|z-z_i|.\] Is it true that $\limsup_{...

Let $p(z)=a_0+a_1z+\ldots+a_nz^n$ be a polynomial, all of whose zeros are on $|z|=1$. If \[A=\max_{0\leq k\leq n}|a_k|,\hspace{1cm}M=\max_{|z|=1}|p(z)...

Let $P_N(z)$ be a polynomial with $N$ terms, satisfying $|P_N(z)|\leq1$ on $|z|=1$. How large can $P_n(z)$ be if $P_n(z)$ is a partial sum of $P_N(z)$...

Is there a function $f(k)$ of the positive integer $k$, so that the square of every polynomial having at least $f(k)$ terms has a least $k$ terms? Erd...

Let $P(z)$ be a polynomial whose zeros $z_1, z_2, \ldots, z_n$ lie in $|z|\leq1$. Is it true that $P'(z)$ always has a zero in $|z-z_1|\leq1$? (Bl. Se...

If $H_\nu(z)$ is the $\nu$-th Hermite polynomial, so that \[H_\nu(z)e^{-z^2}=(-1)^\nu\big(\frac{d}{dz}\big)^\nu e^{-z^2},\] is it true that the equati...

Let $f(z)=z^n+a_1z^{n-1}+\ldots+a_n$ be a polynomial of degree $n$. Cartan proved that the set $|f(z)|\leq1$, which we call $E^{(n)}_f$ can always be ...