Mathematics Problem Archive

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...

Let $\Omega$ be a simply-connected domain in $\mathbb{C}$ with at least two boundary points, and let the function $\phi$ map $\Omega$ analytically and...

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

For $-\infty<p<+\infty$ let \[B(p):=\sup\{\beta_f(p):f\text{ conformal map of }\mathbb{D}\text{ into }\mathbb{D}\}\] where \[\beta_f(p)=\limsup_{r\to1...

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...

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}...

Let $A$ denote the class of functions $f(z) = z + a_2z^2 +\ldots$ analytic in $\mathbb{D}$. For $\delta\geq0$ and $T= \{T_k\}^\infty_2$ a sequence of ...

If $f$ in $A$ and $\delta > 0$, define a $\Sigma_\delta(f)$ neighbourhood of $f$ to be \[\Big\{g:g\in A,\big|(g'(z)-f'(z))-\frac{1}{z}(g(z)-f(z))\big|...

Following the notation of Problem 6.111 and 6.112, it is known that, if $|x|\leq\rho\leq1$ and $\gamma=1/(1+\rho)^2$, then \[N_\gamma\Big(\frac{z}{1-x...

Let $\Gamma$ be a regular curve and $f$ an analytic and conformal function in the open unit disc. Does $f^{-1}(\Gamma)$ necessarily have finite length...

Let $\Gamma$ be a rectifiable curve, and let $E$ be a subset of $\Gamma$ having zero length. If $\Omega$ is any simply-connected domain and $z\in\Omeg...

Let $D$ be a domain in $\mathbb{C}$ containing the origin $0$; for $t>0$, let $\Omega_t$ be the component of $D \cap\{|z|\leq t\}$ containing $0$. In ...

Let $G$ be a domain in $\mathbb{C}$ that contains the origin $0$ and is axially-symmetric with respect to the real axis, that is, if a point $z\in G$ ...

If $A^{(p)}_n=\sup_{f\in S_p}|a_n|$ is it true that \[\frac{A^{(p)}_n}{n^{2p-1}}\to K_p,\hspace{1cm}\text{ as }n\to\infty,\] and if so, what is $K_p$?...

Here our counter-example shows that $|a_n|=o(n^{-\frac{1}{2}})$ is best possible for bounded $f(z)$ in $S(p)$....