Mathematics Problem Archive

If $f(z)=\sum^\infty_{n=1}a_nz^n$ is analytic in $\mathbb{D}$ and $\sum^\infty_{n=1}|a_n|<+\infty$, can $f(z)$ map the unit circle $\mathbb{T}$ onto a...

Let $C$ be a closed curve inside the unit circle $\mathbb{T}$. Under what conditions on $C$ does there exist a univalent function $f$ in $\mathbb{D}$ ...

A function $f(z)$ analytic in $\mathbb{D}$ is said to be typically real if $f(z)$ is real, when and only when $z$ is real, see Rogosinski . If $f(z)=z...

If $f(z)=z+\sum^\infty_{n=2}a_nz^n$ is univalent and star-like of order $\frac{1}{2}$ in $\mathbb{D}$, i.e. \[\text{Re}\,\frac{zf'(z)}{f(z)}\geq\frac{...

A related problem concerns upper bounds for $|a_{n+1}|-|a_n|$ when $f(z)$ is mean $p$-valent. Lucas has proved that \[\big||a_{n+1}|-|a_n|\big|=O(n^{j...

If $f(z)=z+\sum^\infty_{n=2}a_nz^n\in S(1)$, prove that on $|z|=r$, \[|f(z)|\leq\frac{r}{(1-r)^2}.\] It is shown by Garabedian and Royden that $f(z)$ ...

Suppose that $p$ is an integer and $f(z)=\sum^\infty_{n=0}a_nz^n$ is $p$-valent in $\mathbb{D}$. It is conjectured by Goodman that \[|a_n|\leq\sum^p_{...

Suppose that $f(z)=\sum^\infty_{n=0}a_nz^n$ is circumferentially mean $p$-valent and $f(z)\neq0$ in $\mathbb{D}$. (This latter condition is a conseque...

Suppose that \[g(z) = z + b_0 + b_1z^{-1} + \ldots\] is univalent in $|z|>1$. Is it true that for each positive $\varepsilon$ we have \[n|b_n|=O(n^\va...

Suppose that $f(z) = z+\sum^\infty_{n=2}a_nz^n$ in $S$ and that \mbox{$P(z) = \sum^n_{k=0}b_kz^k$} is a polynomial of degree at most $n$. Is it true t...

With the above notation $f (z)$ in $S$ if and only if for each pair of numbers $\xi_1, \xi_2$ satisfying $|\xi_1|\leq1$, $|\xi_2|\leq1$, we have \[f(z...

If $f$ in $S$, Baernstein has shown that \[\int^{2\pi}_0|f(re^{i\theta})|^p\,d\theta\leq\int^{2\pi}_0|k(re^{i\theta})|^p\,d\theta,\hspace{1cm}0<r<1,\h...

Duren has shown that if $f(z) = \sum^\infty_{n=0}a_nz^n$ in $S$ and if \[(1 - r )^2f( r ) = \lambda + O\big(( 1 - r )^\delta\big),\hspace{1cm}\text{ a...

Let $S_\alpha$, $0 < \alpha \leq 1$ be the subclass of $S$ of functions $f$ such that $\mathbb{C}\setminus f(\mathbb{D})$ is a single piecewise analyt...

The same questions as in Problem 6.32 can be asked under the alternative hypothesis that $\mathbb{C}\setminus\{f(\mathbb{D})\}$ is a single piecewise ...

A function $f(z) = z + a_2z^2 +\ldots$ analytic in $\mathbb{D}$ is said to belong to Ruscheweyh's class $M$ if the $*$ (i.e. Hadamard) convolution of ...

Let $\mathbb{O}$ be a subset of $\mathbb{D}=\{|\omega|< 1\}$. Find a characterisation of those $\mathbb{O}$ that are of the form $(\mathbb{C}\setminus...

Suppose that $f$ in $S$ and define \[f_p(z)=[f(z)]^p=z^p+\sum^\infty_{n=p+1}a_{n, p}z^n.\] What can be said about bounds for $a_{n,p}$? If $|a_{n,1}|\...

Suppose that $f(z)=z + c_3z^3 + c_5z^5 +\ldots$ is an odd univalent function in $\mathbb{D}$, and let $d_n = |c_{2n+1}|-|c_{2n-1}|$. It is known that ...

With the notation of Problem 6.37, is it true that \[\sum^\infty_{n=1}n^{-\beta}d_n^2<\infty\] where $\beta=(\sqrt{2}-1)^2$? (K. W. Lucas)...

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 $f(z)$ 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...