Mathematics Problem Archive

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...

(Edrei's spread conjecture) If $f(z)$ is meromorphic in the plane and of lower order $\lambda$, and if $\delta=\delta(a,f)>0$, is it true that, for a ...

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{...

(Paley's conjecture) For any entire function $f(z)$ of finite order $\rho$ in the plane, we have \[1\leq\liminf_{r\to\infty}\frac{\log M(r,f)}{T(r,f)}...

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...

The analogue of Problem 1.7 may be asked for meromorphic functions. The proposers conjecture that in this case \[\sum\delta(a,f)\leq\max\{\Lambda_1(\r...

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 meromorphic in $\mathbb{C}$, and let $f^{-1}$ denote any element of the inverse function that is analytic in a neighbourhood of a point $w$...

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...

Determine the upper and lower estimates for the growth of entire and meromorphic solutions of algebraic ordinary differential equations (AODE). (This ...

Let $F$ be a polynomial in two variables, and let $y$ be a meromorphic solution of the algebraic ordinary differential equation $F(y^{(n)},y)=0$. Is i...

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 meromorphic in $\mathbb{C}$, and suppose that the function \[F(z)=f^{(k)}(z)+\sum^{k-2}_{j=0}a_j(z)f^{(j)}(z)\] is non-constant, where $k\g...

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 $E$...

Let $f(z)$ be an entire function. Then Boas (unpublished) proved that there exists a path $\Gamma_\infty$ such that, for every $n$, $$ \left|\frac{f(z...

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 the entire function $f(z)$ has finite order $\rho$, and the maximal density of non-zero coefficients is $\Delta$, is it true that if $\rho\Delta<\f...

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 $f(z)$ is an entire function, the iterates $f_n(z), n=1,2,\ldots$ are defined inductively by \[f_{n+1}(z)=f(f_n(z)),\hspace{1cm}f_1(z)=f(z).\] A po...

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 $|{f_n}'(z_0)|>1$...

Baker has proved that if $f(z)$ is a transcendental entire function, then $\mathcal{F}(f)$ is not restricted to a straight line in the plane. This imp...

Can an entire function have all its zeros and ones on two distinct straight lines, having infinitely many on each line? Edrei has proved (unpublished)...