Mathematics Problem Archive

If $f(z)$ in $S$ and if the $a_n$ are real, then $$ 1+a_3+\ldots+a_{2n-1}\geq a_n^2,\hspace{1cm}n\geq1. $$ The Bieberbach conjecture for such function...

Let $K(\alpha)$ and $S^*(\alpha)$ be those subsets of $S$ consisting of the class of functions convex in $\mathbb{D}$ of order $\alpha$ i.e. \[\text{R...

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 ...

Using the notation of Problem 6.42, it is well-known that \[\Big|\sum^\infty_{k=1}k\gamma_kz^k\Big|=O\Big(\frac{1}{1-r}\Big),\hspace{1cm}r\to1-,\] for...

Let $f$, $g$ be formal power series \[\sum^\infty_{n=0}a_nz^n,\hspace{1cm} \sum^\infty_{n=0}b_nz^n\] respectively, and define \[(f\otimes g)(z)=\sum^\...

Let $S^*(\alpha)$ be the class of $\alpha$-strongly-star-like functions $f$, that is, those $f$ in $S$ for which \[\Big|\arg\Big(\frac{zf'(z)}{f(z)}\B...

Suppose that $f$ in $S$ and is star-like. Is it true that $$ \big||a_{n+1}|-|a_n|\big|\leq1? $$ This is certainly true if $\lim_{r\to1} (1-r)M(r,f) > ...

If $f$ in $S$ and $f'$ is also univalent in $\mathbb{D}$, what can be said about $\max|a_n|$, $n \geq 2$? The function $z(1-z)^{-1}$ shows that $\max|...

Suppose that $f$ in $S$. The coefficient problem, except in certain cases, remains open for each of the following subclasses of univalent functions. (...

What are the extreme points of the following classes of functions? [(a)] ; Basilevi\^c functions (see Problem 6.48). ; $S^*(\alpha)$ (see Problem 6.48...

If $0 \le \alpha \le 1$, and $f(z)$, $g(z)\in \Sigma$, and if we define $F(z)$ by \begin{eqnarray} F(z)&=&f(z)^{1-\alpha}g(z)^\alpha, \hspace{1cm}|z|>...

Let $D$ be a domain in $\mathbb{C}$ (containing the origin) of connectivity $n$, and let $S(D)$ be the class of analytic univalent functions in $D$ wi...

Suppose that $f(z)$ is analytic in $\mathbb{D}$, and has the whole complex plane as its range. Does there necessarily exist a bounded univalent functi...

Hentgartner and Schobe and Goodman and Saff have shown that if $f(z) = z + a_2z^2 +\ldots$ maps $\mathbb{D}$ univalently onto a domain $G_1$ that is c...

Let $D$ be a Jordan domain with boundary $C$, $\{F_n(z)\}^\infty_1$ the sequence of Faber polynomials for $D$, and $S(D)$ the class of univalent funct...

Let $f(z)$ be a normalised bounded star-like function in $\mathbb{D}$, and set \[f(\xi)=\lim_{r\to1-}f(r\xi),\] where $|\xi|=1$, $\xi\in E$, $E\subset...

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...