Mathematics Problem Archive

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 \[g(z) = z + b_0 + b_1z^{-1} + \ldots\] 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...

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)...

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...

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...

Suppose that $f(z)$ is analytic in $\mathbb{D}$, and has the whole complex plane as its range. Does there necessarily exist a bounded univalent functi...

Hentgartner and Schobe and Goodman and Saff have shown that if $f(z) = z + a_2z^2 +\ldots$ maps $\mathbb{D}$ univalently onto a domain $G_1$ that is c...

Let $D$ be a Jordan domain with boundary $C$, $\{F_n(z)\}^\infty_1$ the sequence of Faber polynomials for $D$, and $S(D)$ the class of univalent funct...

Let $f(z)$ be a normalised bounded star-like function in $\mathbb{D}$, and set \[f(\xi)=\lim_{r\to1-}f(r\xi),\] where $|\xi|=1$, $\xi\in E$, $E\subset...

Let $S_R(q)$ be the class of normalised univalent functions in $\mathbb{D}$ with real coefficients that admit a quasi-conformal extension to the whole...

Following the notation in Problem 6.57, the well-known Golusin inequality for functions $f$ in $\Sigma(q)$ (defined in Problem 6.57) is: $$ \Big|\log\...

Let $D$ be a plane domain containing $\infty$. Let there be given a continuous assignment of numbers (thought of as angles) to the components of $\mat...

Let $C$ be a closed Jordan curve. Then if $f(z) = z + a_2z^2 +\ldots$, $g(z) =z^{-1}+b_0 +b_1z + \ldots$ map $\mathbb{D}$ onto the inside and outside ...

Let $D_1, D_2$ be Jordan domains bounded by rectifiable curves $C_1, C_2$ of equal length. Suppose that an isometric sewing of $C_1$ and $C_2$ is ever...

Let $D_1$ and $D_2$ be bounded Jordan domains, bounded by curves $C_1$ and $C_2$ of bounded boundary rotation (in the sense of Paatero, see e.g. Noona...

Let $\alpha$ be a homeomorphic mapping of $(0, \infty)$ onto $(\alpha(0), \infty)$, $\alpha(0)\geq 0$, such that $x\to\alpha(x)+i$ defines a conformal...

Let $\alpha$ be real and suppose that $f(z)=z+\sum^\infty_{n=2}a_nz^n$ is analytic in $\mathbb{D}$ with $f(z)f'(z)/z\neq0$. We say $f$ is in $M_\alpha...