Mathematics Problem Archive

Let $E_\rho$ be the linear space of entire functions $f$ such that \mbox{$|f(z)|\leq B\exp(A|z|^\rho)$} for some positive $A$ and $B$. Let $K_\rho$ be...

If $A, B$ are countable dense subsets of $\mathbb{R}$, $\mathbb{C}$ respectively, does there necessarily exist a transcendental entire function that m...

If $f(z)$ is a transcendental entire function, we define \[M=\{z:|f(z)|=M(|z|,f)\}.\] Tyler has shown that $M$ can have isolated points, and that, giv...

Characterise those entire functions having at least one continuous maximum modulus path going from $0$ to $\infty$. (W. Al-Katifi)...

Suppose that an entire function $f$ has exactly one curve $\Gamma$ of maximum modulus (that is, $\Gamma$ is connected, joins $0$ to $\infty$, and $f$ ...

What is the best function $g(\sigma)$, $\sigma\geq 0$ such that, for a non-constant entire function $f(z)$ with maximum and minimum modulus $M(r,f)$ a...

For entire or, more generally, meromorphic functions $f$ and $g$, let `$f\leq g$' mean that, for any sequence $\{z_n\}^\infty_1$ for which $|f(z_n)|\t...

Let $E$ be a closed set in $\mathbb{C}$, with the following properties: $(1)$ there exists a transcendental entire function $f(z)$ that is bounded on ...

Let $f_i(z)$, $i=1, 2, 3$ be non-constant entire functions of one complex variable, and \[V=\{z:z=(z_1,z_2,z_3)\in\mathbb{C}^3,f_1(z_1)+f_2(z_2)+f_3(z...

Prove or disprove the conjecture that an entire function $f$ of $n$ complex variables is an $L$-atom (where this is defined in a way analogous to the ...

If $f$ is an entire function such that $\log M(r,f)=O(\log r)^2$ as $r\to\infty$, then Hayman has shown that $\log |f(re^{i\theta})|\sim\log M(r,f)$, ...

Suppose that $f$ is entire with a non-zero Picard exceptional value $\alpha$. Then $f$ has $\alpha$ as an asymptotic value. It can be shown that $f\to...

(A width conjecture) Given a power series $\sum^\infty_{k=0}a_kz^k$, suppose that there is a non-negative $\rho$ such that all of the partial sums $S_...

Let $\sum^\infty_{k=0}a_kz^k$ be a non-vanishing entire function, and let \mbox{$S_n(z)=\sum^n_{k=0}a_kz^k$}. Given $\varepsilon>0$, must there exist ...

Let $\Gamma$ be a rectifiable curve. Suppose $f$ is a continuous function on the plane satisfying \[\int_{\sigma(\Gamma)}f(z)dz=0\hspace{1cm}\text{ fo...

Let $f$ denote a rational or entire function of a complex variable, and $f^n, n=1, 2, \ldots$, the $n$-th iterate of $f$, so that $f^1=f, f^{n+1}=f\ci...

Let $f$ be a rational function and $C$ be as in Problem 2.62. We say that $g$ is a limit function for $f$ if $g$ is defined in some component $G$ of $...

Since the knowledge of the zeros of an entire function $f$ leaves an unknown factor, $e^h$ say, in the Hadamard product for $f$, one can ask if $f$ is...

Given a countable number of entire functions, one can find an entire function growing faster than any of these. Without making any assumption about th...

Let $f$ be an entire function, and let $D$ be a component of the set in $\mathbb{C}$ where the family of iterates $\{f_n\}$ is normal. Can this family...

Let $f$ be an entire function satisfying the condition \[\log M(r,f)\leq(1+o(1))r^\rho,\hspace{1cm}\text{ as }r\to\infty.\] Suppose that there exists ...

Hayman has shown that $$ \liminf_{r\to\infty}\frac{T(r,f)}{T(r,f')}\leq1 $$ for transcendental entire functions $f$ of lower order zero. Toppila has s...

Let $H$ be an entire function, let $f_1, f_2$ be linearly independent solutions of the differential equation $w'' + Hw = 0$, and let $E = f_1 f_2$. Cl...

It is shown by Hellerstein and Rossi , and Gundersen that if $f_1$ and $f_2$ are two linearly independent solutions to the differential equation $w'' ...

Let $\{f_1,\ldots,f_n\}$ be a fundamental system for the differential equation $$ L_n(w)\equiv w^{(n)}+a_{n-1}(z)w^{(n-1)}+\ldots+a_0(z)=0, $$ where $...

Let $F(z, a, b)$ be an entire function of three complex variables, and suppose that $F$ is not of the form $$ F(z,a,b) = G(z,H(a,b)) $$ for any entire...

Suppose that $f(z) = 1 + a_1z + a_2 z^2 +\ldots \in U_{2p}$. If $p = 0$ (so that $f\in U_0)$ and if $f$ is not a polynomial, it is well-known that $f$...

Suppose that $f$ is entire of proximate order $\rho(r)$, and that $f$ has a representation as a Dirichlet series \[f( z ) = \sum^\infty_{n=1}a_ne^{\la...

Let $\Omega$ be a component of the normal set of an entire function (under iteration). Is $\dim(\partial\Omega) > 1$? Or is $\partial\Omega$ a circle/...

Let $\Omega$ be a component of the normal set of an entire function $f$ (under iteration). Do there exist such an $f$ and such an $\Omega$ with the fo...

(Fatou's conjecture) Show that the subset $U$ of functions $g$ in $R_d$, such that all the critical points of $g$ are in the basins of attraction of p...

[(a)] ; Show that, if a function $g$ in $R_d$ has the property that its Julia set $J(g) \neq \hat{\mathbb{C}}$, then $g$ does not leave invariant a no...

Let the function $g$ in $R_d$ have the property that its Julia set $J(g) = \hat{\mathbb{C}}$. Is the dimension $k$ of the space of Beltrami forms on $...

Let the function $g$ in $R_d$ have the property that its Julia set $J(g) = \hat{\mathbb{C}}$. Is $g$ ergodic for Lebesgue measure? In other words, if ...

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 $B$ denote the boundary of the Mandelbrot set (or, equivalently, the topological bifurcation set of the family $z\mapsto z^2+c$, $c\in\mathbb{C}$....

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<+...