31 problems

For every $M>1$, let $f:\mathbb{D}\to\mathbb{C}$ be harmonic and satisfy $|f(z)|<M$ for all $z\in\mathbb{D}$, together with the normalization $f(0)=0$, $f_z(0)=1$, and…

Let $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$. If $f=h+\overline{g}:\mathbb{D}\to\mathbb{C}$ is a normalized sense-preserving harmonic mapping with bounded convex image, where…

For every $n\ge 3$, let $0<r<1$ and let $h:\mathbb{A}(r,1)\to\mathbb{A}(R,1)\subset\mathbb{R}^n$ be an onto homeomorphism whose coordinate functions are harmonic, where…

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…

Clunie–Sheil-Small's coefficient conjecture. For every $n\geq 2$,

Let $f(z)=p(z)\overline{q(z)}$ be a logharmonic polynomial, where $p$ and $q$ are analytic polynomials of degrees $n$ and $m$, respectively, with $p$ not a constant multiple of $q$…

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…

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…

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…

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…

Liu–Ponnusamy's conjectures. (a) If $\varphi$ is univalent and convex in $\mathbb D$, then

Mocanu's conjecture. Every mapping in $\mathcal{M}$ belongs to $\mathcal{S}_{\mathcal{H}}^0$; equivalently, $\mathcal{M}\subset\mathcal{S}_{\mathcal{H}}^0$. The conjecture concerns…

Let $f=h+\overline g\in\mathcal S_H^0$ be a normalized univalent, sense-preserving harmonic mapping, and let $\Omega=f(\mathbb D)$. Theorem 2 in the paper proves the stated spheric…

Ponnusamy–Sairam's conjecture. One has

Let $\mathcal{S}_H^0$ be the subclass of normalized sense-preserving univalent harmonic mappings $f=h+\overline{g}$ with $g'(0)=0$, and let $\mathcal{S}$ be the class of normalized…

Let $f=h+\overline{g}$ be a sense-preserving and univalent harmonic mapping in a simply connected domain $D\subset\mathbb{C}$. Univalent analytic shear conjecture. There exists a c…

Let $A(1,R)$ and $A(1,R_\ast)$ be annuli, and let $\rho$ be the weight defining $(\rho,n)$-harmonicity. For an increasing radial diffeomorphism between them, write … and set…

Equality conjecture.

Let $f=h+\bar{g}\in\mathcal{S}_{H}^{0}$ be a normalized locally univalent harmonic mapping, and let $\Omega=f(\mathbb{D})$ be an $M$-linearly connected domain. Write $\Lambda_f$ fo…