Mathematics Problem Archive

Let $\Gamma\subset\mathbb{C}$ be a Jordan curve of logarithmic capacity $1$, and let $\phi$ be a conformal map from the exterior of $\Gamma$ to the ex...

Let $K$ be a compact subset of $\mathbb{R}^n$, $n\geq3$. For $\phi\in\mathcal{D}$, let $D(\phi)$ be a least-diameter disc containing $\text{spt }\phi$...

Let $f$ be continuous on a compact subset $K$ of $\mathbb{C}$. If there exists a sequence $\{f_n\}^\infty_1$ of functions analytic near $K$ for which ...

Let $f_1$ and $f_2\in H^\infty(\text{unit disc}) = H^\infty$; and let the function $g\in H^\infty$ satisfy the inequality \[|g(z)|\leq|f_1(z)|+|f_2(z)...

Let $\Gamma$ be a curve of the form \[\{x+iA(x):-\infty<x<\infty\}\] with \[|A(x_1)-A(x_2)|\leq M|x_1-x_2|.\] Let $E$ be a compact subset of $\Gamma$,...

Let $K$ denote the $\frac{1}{3}$-Cantor set on $\mathbb{R}$; let $E = K\times K$, and let $\Omega=\mathbb{C}^*\setminus E$. Prove the corona theorem f...

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

For an iteration without memory that uses $n$ evaluations of a function or its derivatives per step, is its convergence order always at most $2^{n-1}$...

Given a complex polynomial $f$ of degree $d\geq2$ and $z\in\mathbb{C}$, must there be a critical point $c$ of $f$ such that $|f(z)-f(c)|\leq |f'(z)|\,...

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

Does the Flint Hills series $\sum_{n=1}^{\infty} 1/(n^3\sin^2 n)$ converge?...

Are there infinitely many Lehmer pairs of zeros in the sense used in the theory of the de Bruijn–Newman constant?...

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...

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=...

Classify the entire solutions of $F(y^{(k)},y)=0$ when $F$ is irreducible and its highest-degree homogeneous part has a single distinct linear factor,...

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...

Let $A$ be a set of vectors $a=(a_1,\ldots,a_n)\in\mathbb C^n$ such that every $n$ of them are linearly independent, and assign to every $a\in A$ a $\...

Let $A$ be a set of vectors $a=(a_1,\ldots,a_n)\in\mathbb C^n$ such that every $n$ are linearly independent, and let $K_a$ be plane convex compact set...

Classify entire functions $f,g$ for which $f\circ g$ is periodic. Prove that, up to the natural equivalences, the possibilities are exhausted by: $g$ ...

Does there exist an entire function whose zeros lie on the positive ray and whose $1$-points lie on two rays making angles $\pm\alpha$ with it, for so...

For $\omega=e^{2\pi i/5}$, is an entire solution of $$f(\omega z)f(\omega^{-1}z)=f(z)-1$$ unique up to rotation of the variable $z$?...

Let $f$ be entire and suppose every zero of every derivative $f^{(n)}$, $n\ge0$, lies in the closed lower half-plane. Must $f$ lie in the compact-open...

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 $\mu$ be a positive plane measure with $\mu(\{|z|\le r\})\le cr^\alpha$ for some $0<\alpha<1/2$. Can its logarithmic potential $$u(z)=\int\log\lef...

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 rational functions of each fixed degree, determine the analogues of Goldberg's constant and the unit-disk zero/one-point extremal quantities, and ...

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...

Obtain better rigorous estimates for the Littlewood exponents $\alpha$, $\beta$, and for $\sup_c P_c$, where $P_c$ is the pressure for the hyperbolic ...

Are the iterates $p_c^n$ of hyperbolic quadratic polynomials extremal, or nearly extremal, for the Littlewood exponent $\alpha$ governing mean spheric...

For which parameters $c$ is the pressure $P_c$ of the hyperbolic quadratic polynomial $p_c(z)=z^2+c$, for the potential $|(p_c^n)'|^{-1}$, close to or...

For a regular connected compact plane set $E$, let $\beta_E=\limsup_{\varepsilon\to0}\log l(\varepsilon)/(-\log\varepsilon)$, where $l(\varepsilon)$ i...

If the definition of $\sup_E\beta_E$ is extended from connected regular compact plane sets to all regular compact sets, is the supremum attained on co...

For finitely many positive charges $a_k$ at points $x_k\in\mathbb R^3$, is the critical set of $u(x)=\sum_{k=1}^n a_k/|x-x_k|$ always finite?...

If the critical set of $u(x)=\sum_{k=1}^n a_k/|x-x_k|$ for positive point charges in $\mathbb R^3$ is finite, how many points can it contain? In parti...

For a monic polynomial $p$ of degree $d$, determine the maximum length of the lemniscate $E(p)=\{z:|p(z)|=1\}$. Is the extremal asymptotically $p(z)=z...

For the rectangular lattice $\Lambda=\{an+ibm:n,m\in\mathbb Z\}$ with $a^2+b^2=1$ and $a\in(0,1)$, let $f_a:\mathbb D\to\mathbb C\setminus\Lambda$ be ...

Let $u_1,u_2,u_3$ be subharmonic in the plane with $u_j(0)=0$, and let $v_1\le v_2\le v_3$ be their pointwise increasing rearrangement. Put $I(r,v)=\i...

Let $f:\mathbb C\to\mathbb P^2$ be linearly nondegenerate and let $\delta(a,f)$ be the Nevanlinna deficiency of a point $a\in\mathbb P^2$. Prove that ...

Let $f:\mathbb C\to\mathbb P^n$ be holomorphic with spherical derivative $\|f'\|(z)=O(|z|^\sigma)$ for some $\sigma>-1$, and let $a_1,\ldots,a_q$ be h...

For $p\ge3$, let $V(D)$ consist of zero-free holomorphic vectors $(f_1,\ldots,f_p)$ on $D$ with $\sum f_j=0$, and use the source's definition of a $C$...