Mathematics Problem Archive

[(a)] ; Suppose that $f, f_1, f_2,\ldots,f_n\in H^\infty$, and \[|f|\leq|f_1|+|f_2|+\ldots+|f_n|.\] Do there necessarily exist $h_1,h_2,\ldots,h_n\in ...

For each pair of $f,g\in H^\infty$, does there necessarily exist another pair of functions $a,b\in H^\infty$ such that $$ af+gb\neq0,\hspace{1cm}|z|<1...

Let $K_1, K_2, K_3$ be disjoint closed sets in the extended complex plane, and $C_1, C_2, C_3$ constants. Let $\rho_n(f)$ be the best rational approxi...

Let $D$ be an open subset of the extended complex plane with non-empty boundary $\partial D$, and let $F$ be a relatively-closed subset of $D$. Let $f...

Let us call a closed set $E$ in $\mathbb{C}$ a weak Arakelian set if, corresponding to each function $g(z)$ continuous on $E$ and analytic in the inte...

Let $\gamma$ be a Jordan arc in $\mathbb{C}^n$, $n\geq2$ such that the projections $\gamma_j$ on the complex coordinate planes $j=1,\ldots,n$ have are...

Does the condition $\sum 1/p_n<\infty$ for positive integers $p_n$ guarantee that the sequences of powers $\{z^{p_n}\}$ fails to span $C(\gamma)$ for ...

Let $F$ be a closed subest of $\mathbb{R}^n$, $n\geq2$. Call $F$ a set of harmonic approximation if every function continuous on $F$ and harmonic in t...

Let $D$ be a planar domain. A sequence $\{z_j\}$ of points in $D$ is said to be an interpolating sequence if whenever $\{\alpha_j\}\in\ell^\infty$ the...

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 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)|\,...

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...

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 $A$ 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...

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...

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 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?...

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 $C$...