Mathematics Problem Archive

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 ...

Assume that $E^{(n)}_f$ is connected. Is it true that $$ \max_{z\in E^{(n)}_f}|f'(z)|\leq\frac{1}{2}n^2\,? $$ Pommerenke proved this with $\frac{1}{2}...

Is it true that to every positive $c$, there exists an $A(c)$ independent of $n$, such that $E_f^{(n)}$ can have at most $A(c)$ components of diameter...

Is it true that the length of the curve $|f_n(z)|=1$ is maximal for $f_n(z)=z^n-1$? (P. Erd\"os)...

If $|z_i|\leq1$, estimate from below, the area of $E^{(n)}_f$. Erd\"os, Herzog and Piranian prove that, given positive $\varepsilon$, the area of $E^{...

It is known that there exists a polynomial $P(z)$ \[P(z)=\sum^n_{k=1}\varepsilon_k z^k, \hspace{1cm}\varepsilon_k=\mp1\] for which $$ \max_{|z|=1}|P(z...

Does there exist a polynomial of the type in Problem 4.13, for which $$ \min_{|z|=1}|P(z)|>C_2\sqrt{n} $$ for every $n$? More generally, does there ex...

If again $\varepsilon_k=\mp1$, is it true that, for large $n$, all but $o(2^n)$ polynomials $P(z)=\sum^n_{k=1}\varepsilon_k z^k$ have just $n/2+o(n)$ ...

Is it true that for all but $o(2^n)$ polynomials $P(z)$ \[\min_{|z|=1}|P(z)|<1,\] or, if not, what is the corresponding correct result?...

If $f$ is any polynomial or rational function of degree $N$, find the least upper bound $\phi(N)$ of \[\frac{1}{r}\int^r_0dt\int^\pi_{-\pi}\frac{|f'(r...

Littlewood conjectured that if $n_1, n_2, \ldots, n_k$ are distinct positive integers then $$ \int^{2\pi}_0\Big|\sum^k_{i=1}\cos (n_1 x)\Big|\,dx>c\lo...

If $a_k=\mp1, k=0,\ldots,n$ and \[b_k=a_na_{n-k}+a_{n-1}a_{n-k-1}+\ldots+a_ka_0,\] is it true that \[\sum^n_1|b_k|^2>An^2,\] where $A$ is an absolute ...

Using the notation of Problem 4.7, if $|z_i|\leq1$, Clunie and Netanyahu (personal communication) showed that a path exists joining the origin to $| z...

Some of the Problems 4.7 to 4.12 extend naturally to the space of higher dimensions. Let $x_i$ be a set of $n$ points in $\mathbb{R}^m$ and let $E^{(m...

Let \[P(z)=\sum^n_0a_kz^k\] be a self-inversive polynomial, i.e. if $\zeta$ is a zero of $P(\zeta)$ with multiplicity $m$, then $1/\zeta$ is also a ze...

Determine \[\inf\int^\pi_{-\pi}\big|1-e^{i\theta}\big|^{2\lambda}\big|P(e^{i\theta})\big|^2\,d\theta,\hspace{1cm}\lambda>0,\] where $P(z)$ ranges over...