Mathematics Problem Archive

Conjecture Any $C^r$ structurally stable diffeomorphism is hyperbolic....

Problem Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?...

Question Give a necessary and sufficient criterion for the sequence $(a_n)$ so that the power series $\sum_{n=0}^{\infty} a_n x^n$ is bounded for all ...

Conjecture Let $D$ be the open unit disk in the complex plane and let $U_1,\dots,U_n$ be open sets such that $\bigcup_{j=1}^nU_j=D\setminus\{0\}$. Sup...

Conjecture For all $\alpha=(\alpha_1,\alpha_2,\ldots)\in l_2(\cal{C})$ the following inequality holds $$\sum_{n\geq 1}|\alpha_n|^2\geq \frac{6}{\pi^2}...

How big can the set of Valiron deficiencies be for functions in the plane? It is known that $$ N(r,a)=T(r,f)+O\big(T(r,f)^{\frac{1}{2}+\varepsilon}\bi...

Let $f(z)$ be an entire function of finite order $\rho$, and let $n_1(r,a)$ denote the number of simple zeros of the equation $f(z)=a$. If \[n_1(r,a)=...

Under what conditions can $\sum\delta(a,f)$ be nearly $2$ for an entire function of finite order $\rho$? Pfluger proved that if $\sum\delta(a,f)=2$, t...

Arakelyan has proved that, given $\rho>\frac{1}{2}$ and a countable set $E$, there exists an entire function $f(z)$ of order $\rho$, for which all the...

If $f(z)$ is an entire function of finite order $\rho$ which is not an integer, it is known that (see Pfluger and Hayman ), \[\sum \delta(a,f)\leq 2-K...

Following the notation in Problem 1.7, if $f(z)$ is meromorphic in the plane of order $\rho$, it is conjectured by Pfluger , that for $a\neq b$ \[\lim...

If $f(z)$ is a meromorphic function of finite order with more than two deficient values, is it true that if $\sigma>1$, then \[\limsup_{r\to\infty}\fr...

If $f(z)$ is a meromorphic function of finite order with at least one finite deficient value, does the conclusion of Problem 1.10 hold?...

Edrei, Fuchs and Hellerstein ask if $f(z)$ is an entire function of infinite order with real zeros, is $\delta(0,f)>0$? More generally, is $\delta(0,f...

If $f(z)$ is an entire function of finite order $\rho$ and lower order $\lambda$ with real zeros, find the best possible bound $B=B(\rho,\lambda)$ suc...

For any function $f(z)$ in the plane, let $n(r)=\sup_a n(r,a)$ be the maximum number of roots of the equation $f(z)=a$ in $|z|<r$, and \[A(r) = \frac{...

If $f(z)$ is non-constant in the plane, it is known (see Hayman ) that \[ \alpha_f=\limsup_{r\to\infty}\frac{T(r,f)}{T(r,f')}\geq \begin{cases} \frac{...

The defect relation ([source label: 1.2]) is a consequence of the inequality (see Hayman ), which is called the ``second fundamental theorem'', $$ \su...

Under what circumstances does $f(z_0+z)$ have the same deficiencies as $f(z)$? It was shown by Dugu{\'e} that this need not be the case for meromorphi...

If $f$ is meromorphic in the plane, can $n(r,a)$ be compared in general with its average value \[A(r)=\frac{1}{\pi}\int\int_{|z|<r}\frac{|f'(z)|^2}{(1...

In the opposite direction to Problem 1.24, does there exist a meromorphic function such that for every pair of distinct values $a, b$, we have \[\lims...

Let $E$ be the set for which $m(r,a)\to\infty$ as $r\to\infty$. How large can $E$ be if: [(a)] ; $f$ is entire and of order $\frac{1}{2}$ mean type, ;...

Are there upper bounds of any kind on the set of asymptotic values of a meromorphic function of finite order? (D. Drasin and A. Weitsman)...

Can one establish an upper bound on the number of finite asymptotic values of a meromorphic function $f(z)$ in $\mathbb{C}$, taking into account both ...

Let the function $f$ be meromorphic in the plane, and not rational, and satisfy the condition $$ \frac{T(r,f)}{(\log r)^3}\to\infty,\hspace{1cm}\text{...

Let $f$ be a meromorphic function of finite order $\rho$. Does the condition \[N(r,1/f')+2N(r,f)-N(r,f')=o(T(r,f)),\hspace{1cm}\text{ as }r\to\infty,\...

Let $n_1(r,a,f)$ denote the number of simple zeros of $f(z)-a$ in $\{|z|\leq r\}$. Selberg has shown that if: [(a)] ; $f$ is a meromorphic function of...

Find criteria for and/or give explicit methods for the construction of meromorphic functions $f$ in $\mathbb{C}$ with the following properties: [(a)] ...

[(a)] ; Let $f$ be non-constant and meromorphic in the open unit disc $\mathbb{D}$, with $\alpha<+\infty$, and define $$ \alpha = \limsup_{r\to1}\frac...

Let $f$ be a function meromorphic in $\mathbb{D}$, for which $\alpha<+\infty$ in ([source label: alphadef]). [(a)] ; Shea and Sons have shown that if ...

Let $f$ be a function meromorphic in $\mathbb{D}$ of finite order $\rho$. Shea and Sons have shown that \[\sum_{a\neq\infty}\delta(a,f)\leq\delta(0,f'...

Let $f$ be a function meromorphic in $\mathbb{D}$, for which $\alpha=+\infty$ in ([source label: alphadef]). Then it is known that \[\sum_{a\in\mathbb...

Let $f$ be a meromorphic function of lower order $\lambda$. Let \[m_0(r,f)=\inf\{|f(z)|:|z|=r\}\] and \[M(r,f)=\sup\{|f(z)|:|z|=r\}\] and suppose that...

Produce a general method for constructing an entire function of finite order, and in fact, minimal growth, which tends to different asymptotic values ...

If $\phi(z)$ is an entire function growing slowly compared with the function $f(z)$, we can consider $\phi(z)$ to be an asymptotic function of $f(z)$,...

Suppose that $f(z)$ is a meromorphic function in the plane, and that for some $\theta$, $0\leq\theta<2\pi$, $f(z)$ assumes every value infinitely ofte...

What can we say about the set $E$ of values $a$ which an entire function $f(z)$ assumes infinitely often in every angle? Simple examples show that $E$...

If $f(z)$ of finite order, can anything be asserted about the length of $\Gamma_\infty$, which is the path on which $f(z)$ tends to $\infty$, or the p...

Does ([source label: 2.1]) remain true if the number $n(r)$ of poles of $f(z)$ in $|z|<r$ satisfies $n(r)=O(r^k)$, where $k<\frac{1}{2}<\lambda$, and ...

We ask the analogues of Problems 2.6, 2.7 and 2.8 if, in addition, $f(z)$ has another finite Picard value, e.g. $f(z)\neq0$. In this case, if $\infty$...

If $f(z)=\sum a_nz^{\lambda_n}$ is an entire function, and $\sum(1/\lambda_n)$ converges, is it true that: [(a)] ; $f(z)$ has no finite asymptotic val...

If $f(z)=\sum a_n z^{\lambda_n}$ is an entire function, and $\lambda_n/n\to\infty$, is it true that $f(z)$ has [(a)] ; no Picard value, ; no Borel exc...

[(a)] ; Let $f(z)=\sum a_n z^n$ be entire and $m(r)=\max_n |a_n|r^n$. If $C>\frac{1}{2}$ then does there exist an entire $f$ with \[m(r)/M(r,f)\to C ?...

(Blumenthal's conjecture) Let $w=f_1(z), f_2(z)$ be entire functions. Is it true that if \[M(r,f_1)=M(r,f_2),\hspace{1cm}0<r<\infty,\] then $f_1(z), f...

Let $\nu(r)$ be the number of points on $|z|=r$, such that \mbox{$|f(z)|=M(r,f)$}. Can we have [(a)] ; $\limsup_{r\to\infty}\nu(r)=\infty$\,? ; $\limi...

If $f(z)$ is a non-constant entire function and \[b(r)=\left(r\frac{d}{dr}\right)^2\log M(r,f),\] then $$ \limsup_{r\to\infty} b(r)\geq A $$ where $A$...

Consider the function $b(r)$ of Problem 2.17. Since $\log M(r,f)$ is an analytic function of $r$, except for isolated points, $b(r)$ exists except at ...

If $f(z)$ is an entire function of exponential type, i.e. satisfying \mbox{$|f(z)|\leq Me^{K|z|}$} for some constants $M$, $K$, and if, further, $|f(x...

If, in the terminology of Problem 2.20, $z_0$ is a fixed point of exact order $n$ for $f(z)$, the fixed point is called repelling if $|{f_n}'(z_0)|>1$...

With the terminology of Problem 2.20, denote by $\mathcal{F}(f)$ the set of points where the sequence $\{f_n(z)\}$ is not normal. Fatou asks if there ...