Mathematics Problem Archive

If $f(z)$ is meromorphic of finite order $\rho$ and $\sum\delta(a,f)=2$, it is conjectured that $\rho=n/2$, where $n$ is an integer and $n\geq 2$, and...

(Edrei's spread conjecture) If $f(z)$ is meromorphic in the plane and of lower order $\lambda$, and if $\delta=\delta(a,f)>0$, is it true that, for a ...

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...

The Bieberbach conjecture Is it true that $|a_n|\leq n$ for $f$ in $S$ with equality only for $f(z)\equiv f_\theta(z)$? The result is known to be true...

Define $A_n=\sup_{f\in S}|a_n|$. It is shown by Hayman that \[\frac{A_n}{n}\to K_0,\hspace{1cm}\text{ as }n\to\infty.\] Is it true that $K_0=1$? The b...

(Schoenberg's conjecture) If $f(z)=\sum^\infty_{n=1}a_nz^n$ and $g(z)=\sum^\infty_{n=1}b_nz^n$ are convex, and $f$, $g$ belong to $S$, is it true that...

Suppose $f$ in $S$ and define $h(z) = \{f(z^2)\}^{\frac{1}{2}} = z + c_3z^3 + c_5z^5 +\ldots$ Robertson's conjecture (see Sheil-Small ) asserts that \...

If $f$ in $S$, write \[\log[f(z)/z]=2\sum^\infty_{k=1}\gamma_kz^k.\] If $f$ is star-like then $|\gamma_k| \leq1/k$; this is false in general, even in ...

It was proved by Boyarski\u\i\, that the partial derivatives of a plane $K$-quasiconformal mapping are locally $L$-integrable for $2\leq p < 2+c$, whe...

This problem is equivalent to Problem 7.9 due to Gehring and Reich about best bounds for area distribution under quasiconformal mapping. Let $E$ denot...

(Subadditivity problem for analytic capacity) Prove or disprove the existence of a constant $M$ such that \[\gamma(K_1\cup K_2)\leq M\{\gamma(K_1)+\ga...

Let $\Gamma$ be a Fuchsian group in $\mathbb{D}$, and let $i(z) \equiv z$. Is it true that \[\sum_{\gamma\in\Gamma}|\gamma'(0)|\geq\prod_{\gamma\in\Ga...

Characterise the Hankel operators on the Hardy space $H^2$ on the circle that are of trace class. (F. Holland)...

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...