55 problems

Let $f=h+\overline{g}\in\mathcal{S}_{\mathcal{H}}$, where $h(z)=z+\sum_{n=2}^{\infty}a_nz^n$ and $g(z)=\sum_{n=1}^{\infty}b_nz^n$ are the analytic and co-analytic parts of a normal…

Let $A_1$ and $A_2$ be annuli with inner and outer radii encoded by $r$ and $R$. Nitsche's conjecture. A harmonic homeomorphism between $A_1$ and $A_2$ exists if and only if … The…

Mocanu's conjecture. Then the harmonic function $f=h+\overline{g}$ is univalent in $\mathbb{D}$. This conjecture concerns a sufficient condition for univalence of a sense-preservin…

Let $p\in[1,\infty]$ and let $q$ be its conjugate. For $\Omega=\mathbb{B}^{n}$ or $\mathbb{H}^{n}$, suppose that $u=\mathcal{P}_{\Omega}[\phi]$ with…

Kalaj's $\rho$-Nitsche conjecture. If there exists a $\rho$-harmonic homeomorphism of $A_1$ onto $A_2$, then

Let $f(z)=\sum_{k=1}^{\infty}a_k z^k+\overline{\sum_{k=1}^{\infty}b_kz^k}\in \mathcal{S}_{H}$. Let $R_f$ be the radius of convergence of $f$, let $R_n$ be the radius of univalence…

Let ${\mathbb D}$ be the unit disk, let $f$ be a harmonic mapping in ${\mathbb D}$, and let $\|S_f\|$ denote its Schwarzian norm. A mapping $f$ admits a homeomorphic extension to…

Let ${\mathbb D}$ be the unit disk, let $\sigma_H({\mathbb D})$ denote the harmonic univalence threshold of ${\mathbb D}$, and write $\|\cdot\|$ for the Schwarzian norm on…

Let $f=h+\overline{g}\in\mathcal{S}_{\mathcal{H}}$, where $h(z)=z+\sum_{n=2}^{\infty}a_nz^n$ and $g(z)=\sum_{n=1}^{\infty}b_nz^n$ are the analytic and co-analytic parts of a normal…

Conjectured order formula.

Let $u\in\mathcal S_{(0,\alpha)-H}^0$ be an $\alpha$-harmonic function with the power-series form given by the $(\alpha,\beta)$-harmonic expansion. For each integer $k\geq2$, the…

Let $u\in\mathcal S_{(\alpha,\beta)}^0$ be a univalent $(\alpha,\beta)$-harmonic function with expansion … Here $F$ denotes the Gauss hypergeometric function. Coefficient conjectur…

Let $\mathcal{S}_{H}$ be the family of sense-preserving planar harmonic univalent mappings $f=h+\overline{g}$ in $\mathbb{D}$ satisfying $h(0)=g(0)=h'(0)-1=0$, and let … Assume the…

The harmonic Bieberbach conjecture. For every $f\in\mathcal{S}_{H}$ and every $n\geq2$,

Let $f=h+\overline{g}\in\mathcal{S}^0_H(K)$, with $h$ and $g$ having the series representations specified in the source. Let $A(n,k)$ and $B(n,k)$ be the quantities defined in the…

Covering-radius conjecture. Every function $f\in\mathcal{S}^0_H(K)$ contains the disk

Covering-radius conjecture. The radius satisfies

Let $\mathcal{S}_\mathcal{H}(K)$ be the class of normalized harmonic $K$-quasiconformal mappings, and write $f=h+\overline{g}$. Harmonic K-quasiconformal Hardy-order conjecture. On…

Let $\mathcal{S}^0_\mathcal{H}(K)$ be the class of normalized harmonic $K$-quasiconformal mappings, and write $f=h+\overline{g}$. Second-coefficient conjecture. One has … The bound…

For $K\geq1$, let $f=h+\overline{g}\in\mathcal{S}^0_\mathcal{H}(K)$, where … Set $k=(K-1)/(K+1)$. HQC Bieberbach conjecture. For every $n\geq2$, … where $A(n,k)$ and $B(n,k)$ are t…

Let $\mathcal{S}^0_\mathcal{H}(K)$ be the class of harmonic $K$-quasiconformal mappings in the normalized univalent harmonic class, and let the harmonic $K$-quasiconformal Koebe fu…

Let $\mathcal{S}_\mathcal{H}$ be the class of normalized univalent harmonic mappings, and let $h^{1/3}$ denote the harmonic Hardy space with exponent $1/3$. Duren's conjecture. One…

Let $\mathcal{S}_\mathcal{H}$ be the class of normalized univalent harmonic mappings, and let $\beta:=\sup_{f\in\mathcal{S}_\mathcal{H}}|a_2|$. The order of the associated harmonic…

Let $\mathcal S_H^0$ be the class of univalent harmonic mappings $f=h+\overline{g}$ of the unit disk $\mathbb D$, normalized by $g(0)=h(0)=g'(0)=0$ and $h'(0)=1$. The Koebe radius…

Let $\mathcal{S}_H$ be the family of all univalent, sense-preserving harmonic mappings $f=h+\overline{g}$ in $\mathbb{D}$ normalized by $h(0)=g(0)=h'(0)-1=0$, and let … Let…