Mathematics Problem Archive

[P. Br\"anden, I. Krasikov, B. Sh., hopefully good, see ] A difference operator $T(p(x))=a_0p(x)+a_1p(x-1)+\cdots+a_kp(x-k)$ with constant coefficient...

If $p$ and $q$ are real-rooted polynomials of degree at most $d$ and of mesh $\geq 1$, then so is $p \bullet q$....

[J. Forsg\aa rd, V. Kostov, B. Sh, hopefully good, see ] For an arbitrary sign pattern $\sigma$, the only type of pairs $(pos,neg)$ which can be non-r...

[J. Forsg\aa rd, B. Sh., seems good, see ] Let $f(z) = \sum_{k=0}^n a_k z^k$ be a polynomial with positive coefficients, and consider the related (wei...

[J. Forsg\aa rd, B. Sh., seems good, see ] Let $f(z) = \sum_{k=0}^n a_k z^k$ be a polynomial with positive coefficients. Consider the differences \[ \...

[J. Forsg\aa rd, B. Sh., seems good, see ]] Let $f(z) = \sum_{k=0}^n a_k z^k$ be a polynomial with positive coefficients. Consider the differences \[ ...

[V. Kostov, B. Sh., looks ugly, see ] What additional restrictions besides [source label: eq:1] exist on configurations $\mathcal A_{f}=\{x^{(i)}_{l}\...

[looks ugly] What symbolic sequences can occur for strictly real-rooted polynomials of degree $n$?...

[B. Sh] For any degree $k$ polynomial $p(x)$ with real coefficients, $$ \#_{r}P_{i}(x) \le \min\{{\deg{P_i(x)},k}\}. $$...

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 $A$...