Mathematics Problem Archive

Let $S_R(q)$ be the class of normalised univalent functions in $\mathbb{D}$ with real coefficients that admit a quasi-conformal extension to the whole...

Following the notation in Problem 6.57, the well-known Golusin inequality for functions $f$ in $\Sigma(q)$ (defined in Problem 6.57) is: $$ \Big|\log\...

Let $D$ be a plane domain containing $\infty$. Let there be given a continuous assignment of numbers (thought of as angles) to the components of $\mat...

Let $C$ be a closed Jordan curve. Then if $f(z) = z + a_2z^2 +\ldots$, $g(z) =z^{-1}+b_0 +b_1z + \ldots$ map $\mathbb{D}$ onto the inside and outside ...

Let $D_1, D_2$ be Jordan domains bounded by rectifiable curves $C_1, C_2$ of equal length. Suppose that an isometric sewing of $C_1$ and $C_2$ is ever...

Let $D_1$ and $D_2$ be bounded Jordan domains, bounded by curves $C_1$ and $C_2$ of bounded boundary rotation (in the sense of Paatero, see e.g. Noona...

Let $\alpha$ be a homeomorphic mapping of $(0, \infty)$ onto $(\alpha(0), \infty)$, $\alpha(0)\geq 0$, such that $x\to\alpha(x)+i$ defines a conformal...

Let $\alpha$ be real and suppose that $f(z)=z+\sum^\infty_{n=2}a_nz^n$ is analytic in $\mathbb{D}$ with $f(z)f'(z)/z\neq0$. We say $f$ is in $M_\alpha...

Given $M$, $1<M<\infty$, let $S^*(M)$ be the class of star-like univalent functions $f$ in $\mathbb{D}$ with $f(0)=0$, $f'(0)=1$, and $|f(z)|\leq M$ f...

Describe the extreme points of the class $\Sigma_0$ consisting of all functions $g$ in $\Sigma$ with constant term $b_0=0$. Springer (see Pommerenke )...

Let $f$ be univalent in $\mathbb{D}$ and let $f(\mathbb{D})$ be a Jordan domain. Does the condition $$ \limsup_{|z|\to1}(1-|z|^2)|f''(z)/f'(z)|<2 $$ i...

Let $\Sigma$ be the class of univalent functions in $\{|z|>1\}$ with the usual normalisation $f(z)=z+\sum^\infty_{n=0}b_nz^{-n}$. Let $S_f$ denote the...

Let $B$ be the Banach space of analytic functions $\phi$ in $\{|z|>1\}$ with finite norm \[\|\phi\|:=\sup_{|z|>1}(|z|^2-1)|z\phi(z)|.\] Let $S$ and $T...

Is every extreme point of $S$ a support point? Is every support point an extreme point? (P. L. Duren)...

For each $f$ in $S$, it can be shown that \[\int^{2\pi}_0\Big|\frac{f'(Re^{i\theta})}{f(Re^{i\theta})}\Big|^2d\theta=O\Big(\frac{1}{1-R}\log\frac{1}{1...

Let $\Gamma$ be the analytic arc omitted by a support point of $S$. Must $\Gamma$ have monotonic argument? Must the angle between the radius and tange...

Let $f(z)=z+\sum^\infty_{n=2}a_nz^n$ be in $S$. Is it true that \[\limsup_{n\to\infty}\big||a_{n+1}|-|a_n|\big|\leq1?\] Hamilton has proved that this ...

Suppose $f(z)=z+\sum^\infty_{n=2}a_nz^n$ is univalent and bounded by $M$ in $\mathbb{D}$. Find \[\sup_t \max_{0\leq t\leq 2\pi}|s_n(e^{it})|,\] where ...

Let $\mathcal{P}_n$ be the class of polynomials \[P_n(z)=z+a_2z^2+\ldots+a_nz^n\] univalent in $\mathbb{D}$, and let \[A_m(n)=\max_{\mathcal{P}_n}|a_m...

Let $\mathcal{V}_n$ denote the class of polynomials \[P_n(z)=z+a_2z^2+\ldots+a_nz^n\] analytic and bi-univalent in $\mathbb{D}$ (that is, $P_n$ and $P...

Let $\mathcal{P}_n$ be the class of polynomials \[p_n(z)=z+a_2z^2+\ldots+a_nz^n\] univalent in $\mathbb{D}$. Determine \[\max_{p\in\mathcal{P}_n}\int^...

Suppose that $f$ in $S$. Consider the region $\mathbb{D}(f)$ on the Riemann sphere which is the stereographic projection of the image of the unit disc...

Let $S_k(\infty)$ denote the class of all analytic and univalent functions $f(z)=z+a_2z^2+\ldots$ defined in $\mathbb{D}$ which admit a $k$-quasiconfo...

If $f$ is univalent analytic in $\mathbb{D}$, then it is well known (see Pommerenke ) that both $f$ and its first derivative $f'$ must be normal, whil...

Let $G$ be the set of functions analytic and not univalent in $\mathbb{D}$. Set, for $f\in G$, \[M_f=\sup\{|f'(z)|:|z|<1\},\hspace{1cm}m_f=\inf\{|f'(z...

The above definition of a bi-univalent function is difficult to understand. What is also meant is that the inverse function $f^{-1}$ has an analytic c...

Let $S$ be the usual class of normalised univalent functions in the unit disc $\mathbb{D}$. Characterise those sequences $\{z_n\}$ of points in $\math...

If $f(z)$ in $S$, write \[\log\frac{f(z)}{z}=2\sum^\infty_{n=1}\gamma_nz^n\] and \[f(z^p)^{1/p}=z+\sum^\infty_{n=1}c^{(p)}_nz^{pn+1}\hspace{1cm}(p=1,2...

Each function $f$ in S that maximises $\text{Re}\, \{L(g) :g \in S\}$ for some continuous linear functional $L$ must map the unit disc onto the comple...

Sundberg notes that it is well known fact (see Hayman ) that, for each fixed $z_0$ in $\mathbb{D}$, \[\Big|z_0\frac{f''(z_0)}{f'(z_0)}-\frac{2\rho^2}{...

Let $L_1$, $L_2$ be two complex-valued continuous linear functionals on $H(\mathbb{D})$, the space of all analytic functions on the unit disc $\mathbb...

Let the function $f = z + a_2z^2 + \ldots$ in $S$ map $\mathbb{D}$ onto a domain with finite area $A$. Then Bieberbach's inequality $|a_2|\leq2$ can b...

Let $S^*(\frac{1}{2})$ denote the class of functions $g$ analytic in $\mathbb{D}$ and such that $\text{Re}\, (zg'/g) > \frac{1}{2}$ in $\mathbb{D}$. I...

Let $E$ be a set of positive logarithmic capacity on the unit circle $\mathbb{T}$. Is $E$ necessarily a set of uniqueness for functions univalent in t...

Let $\Omega$ be an arbitrary domain in $\mathbb{C}$. Does there necessarily exist a set $E$ in $\partial\Omega$, of full harmonic measure, with the fo...

If $\mathbb{R}^2_+=\{(x,y)\in\mathbb{R}^2:y>0\}$, suppose that $E\subset\mathbb{R}^2_+$, and let $f:\mathbb{R}^2_+\to B^2$ be analytic and conformal w...

Let the function $f(z) = z + a_2z^2 + \ldots$ map $\mathbb{D}$ univalently onto a domain $\Omega$, and let $F: \Omega\to\mathbb{D}$ denote the inverse...

Determine an intrinsic characterisation for the class $\mathcal{H}$ of functions $h$ analytic in $\mathbb{D}$ that admit a decomposition of the form $...

Goodman conjectured that if $f(z) = \sum^\infty_{n=1}a_nz^n$ is $p$-valent in $\mathbb{D}$, then for each $n> p$, we have \[|a_n|\leq\sum^p_{k=1}\frac...

The coefficients of a $p$-valent function are bounded by some function of its zeros. In particular, let the function \[f(z)=z^q+\sum^\infty_{n=q+1}a_n...

A function $f(z) = z + a_2 z^2 +\ldots$ is said to belong to the class $CV(R_1,R_2)$ if it is univalent and convex in $\mathbb{D}$, and if on $f(\{|z|...

Given two functions $f$, $g$ in the (usual) class $S$, we can form the new functions (arithmetic and geometric mean functions) \[F(z)=\alpha f(z)+\bet...

Let $K$ be a closed set of points in $\mathbb{C}$, and let $F(K)$ denote the family of functions $f$ of the form \[f(z)=\sum^n_{k=1}\frac{A_k}{z-a_k},...

Let $\{v_n\}^\infty_1$ be a sequence of positive integers (which may include $\infty$); the sequence is called a valence sequence if there is a functi...

The function \[k(z)=2\text{Re }\Big(\frac{z+\frac{1}{3}z^3}{(1-z)^3}\Big)=\sum^\infty_{n=1}\frac{1}{3}(2n^2+1)r^n(e^{in\theta}+e^{-in\theta}),\] where...

It is known that, for functions $f$ in $S^0_H$, $\{|w| < \frac{1}{16}\}\subset f(\mathbb{D})$. Prove that the correct value $d$, of the Koebe constant...

What are the convolution multipliers $\phi^*:K_H\to K_H$, where $K_H$ is the subclass of functions $f$ in $S_H$ with convex images $f(\mathbb{D})$? A ...

Let $J$ be a Jordan curve in $\mathbb{C}$ bounding a domain $D$. Suppose that $f:e^{it}\mapsto f(e^{it})$ is a sense-preserving homeomorphism of the u...

Prove that, for $f\in S^0_H$, $$ \big||a_n|-|a_{-n}|\big|\leq n,\hspace{1cm} n=2,3,4,\ldots. $$ (This is a generalisation of the Bieberbach conjecture...

Let $f$ be analytic univalent in $\mathbb{D}$, and consider \[I_\lambda(r,f')=\Big(\frac{1}{2\pi}\int^{2\pi}_0\big|f'(re^{i\theta})\big|^\lambda\,d\th...