22 problems

Let $cuOmega\textbackslashsubset\textbackslashmathbb R^n$ be a domain, let $K\textbackslashge 1$, and let $p\textbackslashge 1$. A mapping…

Let $f_0:\overline{\mathbb{D}}\to\overline{\mathbb{D}}$ be the prescribed boundary homeomorphism, let $1<p<\infty$, and let…

Resolvent starlikeness conjecture. Every mapping $G_\lambda$ is starlike of order at least $\frac12$. Furthermore, the order of starlikeness of $G_\lambda$ tends to $1$ as…

Let $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ belong to the class $\mathcal{S}_{G}^{\ast}$. The coefficients $a_n$ are the Taylor coefficients of $f$. Coefficient-bound conjecture. The…

Let $U \subset \mathbb R^n$ be open with $n \ge 3$, let $f:U\to f(U)\subset\mathbb R^n$ be a non-constant quasiregular mapping, let $K_I(f)$ denote its inner dilatation, and let…

Kumar–Gangania conjecture. If $f\in\mathcal{S}_{\wp}^{*}$, then the sharp bound is

Let $Z$ be the $z$-axis in $\mathbb{R}^3$, and let $G=\mathbb{R}^3\backslash Z$. The point pair function $p_G$ is defined on pairs of points in $G$. Metricity conjecture. The point…

Metricity conjecture. For a constant $\alpha>0$, this function, defined for $x,y\in\mathbb{R}^n\backslash\overline{\mathbb{B}}^n$ and $n\geq2$, is a metric if and only if…

Let $0<\lambda\leq 1$, and let ${\mathcal U}^+(\lambda)$ denote the class of functions with real coefficients considered in the paper. Write $f(z)=z+a_2z^2+a_3z^3+\cdots$. Coeffici…

Let $\mathcal{S}^{*}_{\wp}$ denote the class of cardioid starlike functions associated with $\psi(z)=1+ze^z$, and let $r_0$ denote the Bohr radius for this class. Bohr-radius conje…

Let $f\in\mathcal{G}_{sh}$ be an analytic function of the form given by equation (1), with Taylor coefficients $a_n$. Coefficient bound conjecture. One should have … The preceding…

Let $\mathbb{B}^2$ be the unit disk, and let $s_{\mathbb{B}^2}$ and $w_{\mathbb{B}^2}$ denote the triangular ratio metric and the function defined in the paper, respectively. The s…

Let $\mathbb{B}^n$ be the unit ball in $\mathbb{R}^n$. For $x,y\in\mathbb{B}^n\setminus\{0\}$, define $\widetilde{x}=x(2-|x|)/|x|$ and $\widetilde{y}=y(2-|y|)/|y|$, and set … Also…

Let $G errbracketmathbb{R}^n$ be a domain, and let $pG$ denote the point pair function. A function $d$ is a quasi-metric if there is a constant $cgeq 1$ such that…

Coefficient monotonicity conjecture. If $f_n\in\mathcal{S}_{l,n}^*$, then

Let $\mathcal G_s(K)$ be the class defined in the paper by, and let $\mathcal F_s(g)$ be the class defined by. For a map $f$, write $K_f$ and $K_{f^{-1}}$ for the distortions of…

Hadamard-product convexity conjecture. For any $f_1,f_2\in\Omega$, one has

Let $\Gamma\subset\mathbb{C}$ be a discrete subset, let $f:\mathbb{D}\to\mathbb{C}\setminus\Gamma$ be a universal covering map, let $r(f)$ denote the relevant covering radius, and…

Conjecture. The function $g$ belongs to the class $\mathcal{U}$ in the disc

Let $\mathcal{F}_3$ be the class of functions considered in the paper, and let $R_{\mathcal{SL}}$ and $R_{\mathcal{M}(\beta)}$ denote the radii for the corresponding classes, with…

Väisälä's uniqueness, prolongation and convexity conjectures. There are universal constants $c_u,c_p,c_c>0$ such that:

Let $\Omega\subset\mathbb{R}^n$ be a domain, let $f\in W^{1,n}_{\operatorname{loc}}(\Omega,\mathbb{R}^n)$, and let $K_I(\cdot,f)$ denote the inner distortion of $f$. Assume that…