Mathematics Problem Archive

(Paley's conjecture) For any entire function $f(z)$ of finite order $\rho$ in the plane, we have \[1\leq\liminf_{r\to\infty}\frac{\log M(r,f)}{T(r,f)}...

The analogue of Problem 1.7 may be asked for meromorphic functions. The proposers conjecture that in this case \[\sum\delta(a,f)\leq\max\{\Lambda_1(\r...

Let $f$ be meromorphic in $\mathbb{C}$, and let $f^{-1}$ denote any element of the inverse function that is analytic in a neighbourhood of a point $w$...

Determine the upper and lower estimates for the growth of entire and meromorphic solutions of algebraic ordinary differential equations (AODE). (This ...

Let $F$ be a polynomial in two variables, and let $y$ be a meromorphic solution of the algebraic ordinary differential equation $F(y^{(n)},y)=0$. Is i...

Let $f$ be meromorphic in $\mathbb{C}$, and suppose that the function \[F(z)=f^{(k)}(z)+\sum^{k-2}_{j=0}a_j(z)f^{(j)}(z)\] is non-constant, where $k\g...

Let $f(z)$ be an entire function. Then Boas (unpublished) proved that there exists a path $\Gamma_\infty$ such that, for every $n$, $$ \left|\frac{f(z...

If the entire function $f(z)$ has finite order $\rho$, and the maximal density of non-zero coefficients is $\Delta$, is it true that if $\rho\Delta<\f...

If $f(z)$ is an entire function, the iterates $f_n(z), n=1,2,\ldots$ are defined inductively by \[f_{n+1}(z)=f(f_n(z)),\hspace{1cm}f_1(z)=f(z).\] A po...

We can also compare $m_0(r,f)$ with the characteristic $T(r)$. We have \[\limsup_{r\to\infty}\frac{\log m_0(r,f)}{T(r)}\geq D(\lambda)\] and ask for t...

(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 $\Gamma$ 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 $f$ 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 $...

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 ...

(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 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...

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 $P(z)$ 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...

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 ...

Suppose that $P$ is a non-linear polynomial with real coefficients. Show that $P^2(z)+P'(z)$ has non-real zeros. We conjecture that the lower bound fo...

Let $\mathcal{P}$ denote the set of all polynomials of the form \[p(z)=\prod^n_{\nu=1}(z-\zeta_\nu),\] where $n\geq2$ and $|\zeta_\nu|\leq1$, $\nu=1, ...

(The corona conjecture) Let $D$ be an arbitrary domain in the plane that supports non-constant bounded analytic functions. Suppose that $f_1(z),\ldots...

Suppose that $f(z)=\sum^\infty_{n=0}a_nz^n$ is circumferentially mean $p$-valent and $f(z)\neq0$ in $\mathbb{D}$. (This latter condition is a conseque...

If $f$ in $S$, Baernstein has shown that \[\int^{2\pi}_0|f(re^{i\theta})|^p\,d\theta\leq\int^{2\pi}_0|k(re^{i\theta})|^p\,d\theta,\hspace{1cm}0<r<1,\h...

Let $\Omega$ be a simply-connected domain in the finite plane whose complement contains $n$ disjoint closed balls with centres on the interval $[0,1]$...

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 $A$ 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 $A$ ...

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...

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...

Prove or disprove the following statements about analytic capacity $\gamma$. [(a)] ; If $E\subseteq\mathbb{C}$ is compact and $\phi$ is a $C^1$-diffeo...

Do there exist inner functions in any strictly pseudo-convex domain of $\mathbb{C}^n$, $n\geq2$? Alexandrov (no citation) and independently L{\o}w (no...

Let $\psi:S^1\to S^1$ be a homeomorphism. When is it true that $\text{Re }A = \text{Re }A \circ \psi$? That is, when is each function $f$ in the real ...

For nonconvex subsets of $\mathbb{R}$ and $\mathbb{R}^2$, is a set spectral if and only if it tiles by translations?...

Characterize the domains for which there exists a nonzero function whose integral vanishes over every congruent copy of the domain....

Let $\boldsymbol{d}=(d_1,\ldots,d_n)$ be an unordered partition of $4g-4$ with $d_i\in\{-1,0,1,2,\ldots\}$, and let $\widehat\Pi_{4g-4}$ be the set of...

Let $\boldsymbol{d}=(d_1,\ldots,d_n)$ be an unordered partition of $4g-4$ with $d_i\in\{-1,0,1,2,\ldots\}$, and let $\widehat\Pi_{4g-4}$ be the set of...

Fix singularities $a_1,\ldots,a_n$ and real exponent differences $\alpha_1,\ldots,\alpha_n$ for second-order Fuchsian equations on the Riemann sphere....

For the Heun equation $$y''+\left(\sum_{j=0}^2\frac{1-\alpha_j}{z-a_j}\right)y'+\frac{Az-\lambda}{(z-a_0)(z-a_1)(z-a_2)}y=0,$$ where $\alpha_j>0$, $A=...

Let $D\subset\mathbb R^n$ have regular boundary for the Dirichlet problem. Prove or disprove that the following are equivalent: (PI) there is a nonzer...

For a local inverse germ $\phi_z$ of a meromorphic function $f$ at a noncritical value $w=f(z)$, Gross's theorem gives analytic continuation along alm...

Let $F$ be entire in two variables and let a holomorphic germ $\phi$ satisfy $F(z,\phi(z))=0$ near a nonsingular point. Must $\phi$ admit analytic con...

Let subharmonic functions $u_k$ on the unit square converge uniformly to $u(x,y)=x$. If $D_k=\{u_k<0\}$ and $D_k^*$ is the component containing $-1/2$...

Let $u$ be subharmonic on $1<|z|<2$, and let pairwise disjoint open sets $D_k$ be unions of components of $\{u<0\}$. Suppose that for every $r\in(1,2)...

Let positive masses $a_k$ be placed at a discrete set $x_k\in\mathbb R^n$ and suppose $\sum_k a_k/|x_k|^{n-1}<\infty$, so that $$F(x)=\sum_k\frac{a_k(...

Let $f$ be holomorphic in the unit disk with exactly one simple zero $z_0$ and two $1$-points $z_1,z_2$, counted with multiplicity. Determine the larg...

Let $f$ be holomorphic in the unit disk, with $f(0)=0$, $f'(0)\ne0$, no other zeros, and exactly two solutions $z_1,z_2$ of $f(z)=1$, counted with mul...

Solve the two-one-point extremal problem for real holomorphic $f$: determine the minimum of $\max(|z_1|,|z_2|)$ and maximum of $|f'(0)|$ when $f(0)=0$...

Let $f$ be real and holomorphic in the unit disk, with one simple zero at $0$ and two simple $1$-points at $\pm ia$. Determine the minimum possible va...

For $\phi(n)=\sup_{\deg p=n}\int_{|z|<1}|p'|/(1+|p|^2)\,dm$, let $\alpha=\limsup\log\phi(n)/\log n$. For a regular compact set $E$, define $\beta_E$ f...