Mathematics Problem Archive

It is known that there exist functions which fail to take any of the values $2\pi ik$, $-\infty<k<+\infty$ and which do not satisfy ([source label: 5....

If $c_k$ is a sequence of positive numbers such that \[\sum c_k=S<+\infty,\] and $n_k$ is an arbitrary sequence of positive integers, then \[f(z)=\sum...

Suppose that $f(z)=z+a_2z^2+\ldots$ is analytic in $\mathbb{D}$. Then $f(z)$ maps some sub-domain of $\mathbb{D}$ univalently into a disc of radius at...

With the hypotheses of Problem 5.8, it follows that $f(z)$ assumes all values in some disc of radius $L$, $L\geq B$. What is the value of $L$? The bes...

If, in addition, $f(z)$ is univalent in $\mathbb{D}$, the conclusions of Problem 5.8 and Problem 5.9 follow with a constant $S$, $S\geq L$, known as t...

$f(z)$ meromorphic in $\mathbb{D}$, $f(z)\neq0, f^{(l)}(z)\neq1$, where $l\geq1$....

$f(z)$ meromorphic in $\mathbb{D}$, $f'(z)f(z)^n\neq1$, for $n\geq3$....

$f'-f^n\neq a$, where $a$ is some complex number, and $n\geq 5$ if $f$ is meromorphic, $n\geq3$ if $f$ is entire. The corresponding results for functi...

Is it possible to remove the restriction that $D^*$ is simply connected in $(a)$ and $(b)$ above? It might be possible to start with the case when $D$...

Do corresponding results to Problem 5.15(a) apply to the means \[I_\lambda(r,f)=\Big\{\frac{1}{2\pi}\int^{2\pi}_0\big|f(re^{i\theta})\big|^\lambda \,d...

Let $D=D_0$ be a domain, $g(z,a_0)$ be the Green's function of $D$ with respect to a point $a_0$ on the positive real axis, and let $D_\lambda$ be the...

Let $f(z)=\lambda+a_1z+\ldots$ be analytic in $\mathbb{D}$, where $0<\lambda<1$. Find the best constant $B(\lambda)$ such that if \[F(r)=\lambda+|a_1|...

A function meromorphic in $\mathbb{D}$ which has no asymptotic value, assumes every value infinitely often in the disc. Every point of the circumferen...

Plessner proves after Privaloff that if $f$ is analytic in $\mathbb{D}$, almost all points $P$ of the boundary are of two kinds. Either [(a)] ; $f$ te...

Corresponding to each function $f$ analytic in $\mathbb{D}$, and each value $w$, with $(|w|<1)$, write \[f_w(z)=f\Big(\frac{z-w}{1-wz}\Big)=\sum^\inft...

Let $H^p$ be the space of functions $f(z)=\sum^\infty_{n=0}a_nz^n$ analytic in $\mathbb{D}$, and such that \[\int^{2\pi}_0\big|f(re^{i\theta})\big|^p\...

Describe similarly the coefficient multipliers from $S$ to $S$, where $S$ is the class of functions $\sum^\infty_{n=1} a_nz^n$ univalent in $\mathbb{D...

Is the intersection of two finitely generated ideals in $H^\infty$ finitely generated? (L. A. Rubel)...

Let $W^+$ be the Banach algebra of power series $f(z)=\sum^\infty_{n=0}a_nz^n$ absolutely convergent in $|z|\leq1$, with $\|f\|=\sum^\infty_{n=0}|a_n|...

Let $f$ be continuous in $\overline{\mathbb{D}}$ and analytic in $\mathbb{D}$. Let \[\omega(f,\delta)=\sup|f(z)-f(w)|,\hspace{1cm}\text{for }|z-w|\leq...

A $G_\delta$ set is a subset of a topological space that is a countable intersection of open sets. Let $E$ be a $G_\delta$ set of measure zero on $|z|...

Let $\mathcal{B}$ be the space of Bloch functions, that is the space of functions analytic in $\mathbb{D}$ with \[\|f\|_\mathcal{B}=|f(0)|+\sup_\mathb...

It was shown by Becker that \[\big\{f:\|f\|_\mathcal{B}<1\big\}\subset \mathcal{B}_Q.\] Is the radius 1 best possible? Is it true that for $f \in \mat...

Suppose that $f_n\in\mathcal{B}_S$. What does $\|f_n-f\|_\mathcal{B}\to0$ as $n\to\infty$ mean geometrically for the functions $g_n$ related to $f_n$ ...

Let $L$ be a regular triangular lattice in the plane. Let $f(z)$ map $\mathbb{D}$ onto the universal covering surface over the complement of $L$. Is i...

It was proved by Hall that every Bloch function has (possibly infinite) angular limits on an uncountably dense subset of $|z | = 1$. Do there always e...

Let $F$ be any discontinuous group of M\"obius transformations of $\mathbb{D}$. Does there always exist a meromorphic function automorphic with respec...

Let $(n_k)$ be a sequence of positive integers such that $$ n_{k+1}>\lambda n_k,\text{ where }\lambda>1, $$ and suppose that $$ f(z)=\sum^\infty_{k=0}...

Suppose that $f(z)$ is a function as in ([source label: B5.13]) and define \[\mu=\limsup_{r\to1}\frac{\log\log M(r,f)}{-\log(1-r)},\] where $M(r,f)$ i...

Tao-Shing Shah has shown that if $g(z)\prec f(z)$ in $\mathbb{D}$, $g'(0)/f'(0)$ is real, and $f$ in $S$ then $$ |g(z)|\leq|f(z)|\text{ for }|z|\leq\f...

Goluzin has shown that, if $g(z)\prec f(z)$ in $\mathbb{D}$, then \[M_2(r,g')\leq M_2(r,f'),\hspace{1cm}0\leq r\leq\frac{1}{2}.\] Here, for $p > 0$, \...

Suppose that $f(z) = \sum^\infty_0 a_nz^n$ and that $F(z)$ is analytic in $\mathbb{D}$, with $f\prec F$. What non-trivial conditions on $F$ imply that...

Suppose that \[f(z)=\sum^\infty_{n=0}a_nz^n\] is analytic in $\mathbb{D}$, with \[\sum^\infty_{n=0}|a_n|=1,\hspace{1cm} |f(z)|\geq\delta>0\text{ in }\...

Determine the Laurent coefficient bodies for analytic functions taking values of modulus at most unity in a given annulus \[A_r = \{z : r < |z| < 1\}....

Suppose that $0 < \alpha < 1$ and \[\frac{(1+xz)^\alpha}{1-z}=\sum^\infty_{n=0}A_n(x)z^n,\hspace{1cm} A_0(x)=1,\hspace{1cm} |x|=1.\] Is it true that \...

If $A$ is any analytic subset of the Riemann sphere it was shown by Kierst that $A$ is (exactly) the set of asymptotic values of a function meromorphi...

A non-constant function $f$ analytic in $\mathbb{D}$, is said to be in the MacLane class $\mathcal{A}$ if the set of points of the unit circle $\mathb...

Let \[f(z)=\sum^\infty_{n=0}a_nz^{\lambda_n}\] be analytic in $\mathbb{D}$ and have Hadamard gaps, i.e. $\lambda_{n+1}/\lambda_n\geq q>1$. Need $f$ ha...

Let \[f(z)=\sum^\infty_{k=1}a_{n_k}z^{n_k}\] be analytic in $\mathbb{D}$ and have Hadamard gaps. Characterise sets $S$ in $\mathbb{D}$ with the proper...

What kind of gaps can the Taylor expansion of a non-constant automorphic function have? For example, can it have Hadamard gaps? (Presumably the sharp ...

A function $f$ analytic in $\mathbb{D}$, is said to be annular if there exists a sequence $\{J_n\}$ of Jordan curves in $\mathbb{D}$ such that [(a)] ;...

It can be shown that there exists a Blaschke product $B(z)$ with \mbox{$B(0) = 0$} such that \[\frac{1+B(z)}{1-B(z)}\] is a Bloch function. Give an ex...

Let F be a Fuchsian group in $\mathbb{D}$, and let $B$ be a set on $\mathbb{T}$ such that $B\cap \gamma(B)=\emptyset$ for $\gamma\in\Gamma,\gamma\neq ...

The ratio $R$ of two Blaschke products $B(z,a_n), B(z,b_n)$ is of bounded characteristic, but need not be a normal meromorphic function. That is, if $...

Suppose that \[f(z)=\sum^\infty_{n=0}a_nz^n\] is convergent in $\mathbb{D}$, that $|z_0| = 1$, and that neither of the series \[\sum^\infty_{n=0}(\tex...

Let the function $h(z)$ be analytic in $\mathbb{D}$, $G_h = \{h(\mathbb{D})\}$, $K_h$ a compact subset of $G_h$. The function $h(z)$ will be said to h...

By a first-order property we shall mean a ring-theoretic property in the first-order predicate calculus. For functions $f$ analytic in $\mathbb{D}$, a...

Suppose that $f$ is analytic in $\mathbb{D}$. Plessner's theorem asserts that at almost all points $e^{i\theta}$ on the unit circle, either $f$ has an...

Suppose that $f$ is univalent and zero-free in $\mathbb{D}$. It has been shown by Baernstein that, for each $p\in(0,\frac{1}{2})$, $f$ admits a factor...

(Subordination and extreme point problem) Let $g(z)=\sum^\infty_{n=0}B_nz^n$ be analytic in $\mathbb{D}$. Denote by $S_g$ the family of functions $f(z...