Mathematics Problem Archive

[(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 ...

Define $A_n=\sup_{f\in S}|a_n|$. It is shown by Hayman that \[\frac{A_n}{n}\to K_0,\hspace{1cm}\text{ as }n\to\infty.\] Is it true that $K_0=1$? The b...

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...

(Schoenberg's conjecture) If $f(z)=\sum^\infty_{n=1}a_nz^n$ and $g(z)=\sum^\infty_{n=1}b_nz^n$ are convex, and $f$, $g$ belong to $S$, is it true that...

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 $f(z)=\sum^\infty_{n=0}a_nz^n$ is circumferentially mean $p$-valent and $f(z)\neq0$ in $\mathbb{D}$. (This latter condition is a conseque...

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...

If $f$ in $S$, Baernstein has shown that \[\int^{2\pi}_0|f(re^{i\theta})|^p\,d\theta\leq\int^{2\pi}_0|k(re^{i\theta})|^p\,d\theta,\hspace{1cm}0<r<1,\h...

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}|\...

Suppose that $f(z)=z + c_3z^3 + c_5z^5 +\ldots$ is an odd univalent function in $\mathbb{D}$, and let $d_n = |c_{2n+1}|-|c_{2n-1}|$. It is known that ...

With the notation of Problem 6.37, is it true that \[\sum^\infty_{n=1}n^{-\beta}d_n^2<\infty\] where $\beta=(\sqrt{2}-1)^2$? (K. W. Lucas)...

Suppose $f$ in $S$ and define $h(z) = \{f(z^2)\}^{\frac{1}{2}} = z + c_3z^3 + c_5z^5 +\ldots$ Robertson's conjecture (see Sheil-Small ) asserts that \...

If $f(z)$ in $S$ and if the $a_n$ are real, then $$ 1+a_3+\ldots+a_{2n-1}\geq a_n^2,\hspace{1cm}n\geq1. $$ The Bieberbach conjecture for such function...

Let $K(\alpha)$ and $S^*(\alpha)$ be those subsets of $S$ consisting of the class of functions convex in $\mathbb{D}$ of order $\alpha$ i.e. \[\text{R...

If $f$ in $S$, write \[\log[f(z)/z]=2\sum^\infty_{k=1}\gamma_kz^k.\] If $f$ is star-like then $|\gamma_k| \leq1/k$; this is false in general, even in ...

Using the notation of Problem 6.42, it is well-known that \[\Big|\sum^\infty_{k=1}k\gamma_kz^k\Big|=O\Big(\frac{1}{1-r}\Big),\hspace{1cm}r\to1-,\] for...

Let $f$, $g$ be formal power series \[\sum^\infty_{n=0}a_nz^n,\hspace{1cm} \sum^\infty_{n=0}b_nz^n\] respectively, and define \[(f\otimes g)(z)=\sum^\...

Let $S^*(\alpha)$ be the class of $\alpha$-strongly-star-like functions $f$, that is, those $f$ in $S$ for which \[\Big|\arg\Big(\frac{zf'(z)}{f(z)}\B...

Suppose that $f$ in $S$ and is star-like. Is it true that $$ \big||a_{n+1}|-|a_n|\big|\leq1? $$ This is certainly true if $\lim_{r\to1} (1-r)M(r,f) > ...

If $f$ in $S$ and $f'$ is also univalent in $\mathbb{D}$, what can be said about $\max|a_n|$, $n \geq 2$? The function $z(1-z)^{-1}$ shows that $\max|...

Suppose that $f$ in $S$. The coefficient problem, except in certain cases, remains open for each of the following subclasses of univalent functions. (...

What are the extreme points of the following classes of functions? [(a)] ; Basilevi\^c functions (see Problem 6.48). ; $S^*(\alpha)$ (see Problem 6.48...

If $0 \le \alpha \le 1$, and $f(z)$, $g(z)\in \Sigma$, and if we define $F(z)$ by \begin{eqnarray} F(z)&=&f(z)^{1-\alpha}g(z)^\alpha, \hspace{1cm}|z|>...

Let $D$ be a domain in $\mathbb{C}$ (containing the origin) of connectivity $n$, and let $S(D)$ be the class of analytic univalent functions in $D$ wi...