Mathematics Problem Archive

(Hadamard convolutions) Suppose that $\alpha\geq1, \beta\geq1$ and that $\phi$ is analytic in $\mathbb{D}$, and satisfies \[\phi(z)\ast\frac{(1+xz)^\a...

Let $w(z)$ be analytic in $\mathbb{D}$ with $w(0)=0$. If \mbox{$|w(z)+zw'(z)|<1$}, for $|z|<1$, then a simple application of Schwarz's lemma shows tha...

Let $u$ be a continuous real-valued function on the unit circle $\mathbb{T}$. Give a necessary and sufficient condition on $u$ such that $u$ is the re...

One of the many equivalent norms on BMOA on $\mathbb{D}$ is defined by \[\|f\|_h=\inf_q\sup_{z\in\mathbb{D}}|f(z)+\overline{q(z)}|,\] the infimum bein...

Let $f$ be analytic in $\mathbb{D}$ with $$ |f(z)|=O((1-|z|)^{-k}),\hspace{1cm}k\geq0. $$ Then $f$ induces a distribution on $C^\infty(T)$, as follows...

Does there exist a non-constant function $f$ in the disc algebra such that $f(e^{i\theta})\in f(\mathbb{D})$ for almost all $\theta$? Caution: the Rud...

Let $B$ be an infinite Blaschke product in $\mathbb{D}$. Does there exist a positive $\delta$, depending on $B$, such that, for every $w$, $|w|<\delta...

Let the function $f$ in $\mathbb{D}$ be given by \[f(z)=\sum^\infty_{k=0}a_kz^{n_k},\hspace{1cm}\frac{n_{k+1}}{n_k}\geq\lambda>1,\hspace{1cm} k\geq0,\...

Let the function $f$ where \[f(z)=1+\sum^\infty_{n=1}a_nz^n,\hspace{1cm}|z|\leq1,\] be a Bloch function with positive real part in $\mathbb{D}$. Deter...

Let the function $f$ where \[f(z)=1+\sum^\infty_{n=1}a_nz^n,\hspace{1cm}|z|\leq1,\] be a Bloch function with positive real part in $\mathbb{D}$ and su...

Barth and Clunie have constructed a bounded analytic function in $\mathbb{D}$ with a level set component of infinite length; this component is highly ...

Suppose that \[f(z) = \sum^\infty_{k=1}a_kz^{n_k}, \hspace{1cm}n_{k+1}/n_k\geq q>1,\] is an analytic function in $\mathbb{D}$ with Hadamard gaps, such...

Let the function $f$ have the power series $f(z)=\sum^\infty_{n=0}a_nz^n$ of radius of convergence $1$; let $E$ be the singular set on $\mathbb{T}$, a...

Let $0<\alpha<1$ and let $R_\alpha$ denote the set of all Riesz potentials $p(x)$ of finite positive Borel measures $\mu$ on $\mathbb{R}$: \[p(x)=\int...

Characterise those non-negative measurable functions $f$ on the unit circle that are dominated almost everywhere by moduli of the boundary values of a...

Does there exist a bounded analytic function in $\mathbb{D}$ such that the image of every radius has infinite length? See, for example, Anderson , and...

Let $\gamma$ be a non-tangential arc that lies in $\mathbb{D}$ except for one endpoint at $z = 1$, and define \[\gamma_\theta=e^{i\theta\gamma},\hspac...

In general, the radial behaviour of the derivative of a bounded analytic function in $\mathbb{D}$ can be pretty arbitrary; in fact, even under much st...

Let $H^1$ denote Hausdorff one-dimensional measure on $\mathbb{C}$, and $\mathbb{T}$; let $g:\mathbb{T}\to[-\infty,\infty]$ denote an arbitrary Borel ...

[(a)] ; Let $f$ be a non-constant analytic function in $\mathbb{D}$, $m$ be a positive integer, and define $\psi = (f)^mf'$. Then it is shown by Sons ...

If $f(z)$ in $S$ it is shown by Bombieri , that there exist constants $c_n$ such that for $f(z)$ in $S$ \[|\text{Re}\,(n-a_n)|\leq c_n\text{Re}\,(2-a_...

If it proves too difficult to obtain sharp bounds for all of the coefficients in Problem 6.4, we ask for the orders of magnitude. An area principle sh...

What are the orders of magnitude of the $c_n$ in Problem 6.4? Springer obtained the estimate \[|c_n|\leq\frac{2^n}{n}\] and also showed that, given $\...

If $f(z)$ in $S$ and is bounded, i.e. satisfies $|f(z)|<M$ for $z\in\mathbb{D}$, we again ask for the order of magnitude of the coefficients $a_n$. Si...

We write \[I_\lambda(r,f)=\Big\{\frac{1}{2\pi}\int^{2\pi}_0|f(re^{i\theta}|^\lambda\,d\theta\Big\}^{1/\lambda}.\] What are the exact bounds for $I_\la...

If $F(z)$, $G(z)$ are convex functions in $\Sigma$, it is known that for $0<\lambda<1$, \[H(z)=\lambda F(z)+(1-\lambda)G(z)\in\Sigma,\] see Pommerenke...

If $f(z)$, $g(z)$ are convex functions in $S$, is it true that for $0<\lambda<1$, $\lambda f+(1-\lambda)g$ is star-like and univalent? A function $w=f...

If $f(z)=z+\sum^\infty_{k=2}a_{n_k}z^{n_k}\in S$, and \[\liminf_{k\to\infty}\frac{n_{k+1}}{n_k}>1,\] then Pommerenke has proved that $$ a_n=o\Big(\fra...

Suppose that $f(z)$ in $S$, and that positive integers $k, m, n,$ are given. It is known that there exist complex numbers $c_0, c_1,\ldots,c_m,$ depen...

If $f(z)$ in $S$, set \[A^{(k)}_n= \begin{vmatrix} a_n,&a_{n+1},&\ldots,&a_{n+k-1} \hdotsfor{4} a_{n+k-1},&a_{n+k},&\ldots,&a_{n+2k-2} \end{vmatrix}\]...

If $f(z)$ in $S$, write \[f_\alpha(z)=\int^z_0f'(\zeta)^\alpha\, d\zeta.\] For what values of $\alpha$, is it true that $f_\alpha(z)\in S$? The result...

Let $S^*$ be the class of all star-like functions $f(z)$ in $S$. Marx conjectured that for each fixed $z_0$, $|z_0|<1$, the set of all numbers $f'(z_0...

If $f(z)=z+\sum^\infty_{n=2}a_nz^n$ in $S$, then \[ A=\pi\sum^\infty_{n=1}n|a_n|^2\] is the area of the image domain. What is the minimum value of $A$...

If $F(z)=z+\sum^\infty_{n=1}b_nz^{-n}$ in $\Sigma$, then \[A(F)=\pi-\pi\sum^\infty_{n=1}n|b_n|^2\] is the area of the set of values not assumed by $F(...

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 \[g(z) = z + b_0 + b_1z^{-1} + \ldots\] 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...

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}|\...