93 problems

Let $\mathcal{S}$ be the class of univalent functions of the form … in the unit disk, and let $n\geq 2$. Zalcman's conjecture. One has … Equality is attained by the Koebe function…

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…

Let $\Sigma$ denote the class of bi-univalent functions normalized by $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$. Brannan–Clunie conjecture. Every function $f\in\Sigma$ satisfies … The con…

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…

Ma's generalized Zalcman conjecture. If $f\in\mathcal{S}$, then

Let $k=\|\mu\|_\infty<1$ and let $K$ be the distortion parameter inferred from $k$ through the relation specified in the source. Consider the area-distortion theorem stated immedia…

Let $\mathcal{S}$ denote the class of normalized univalent functions in the unit disk, and let $\mathcal{S}\ast\mathcal{S}$ be the class of convolutions $f\ast g$ with…

Ahlfors–Grunsky conjecture. The upper bound is sharp, namely

Let $\Omega\subset\mathbb{R}^{n}$ be a domain, and let $f\in W_{loc}^{1,n}(\Omega,\mathbb{R}^{n})$ be a Sobolev map with distortion function $K\in L_{loc}^{n-1}(\Omega)$. Iwaniec–Š…

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

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…

Let $f$ and $g$ be convex univalent functions in the unit disk, and let their convolution be defined by coefficientwise multiplication of their Taylor series, … where…

Let $\Omega\subset\mathbb R^n$ be a domain, let $F\in W^{1,n}_{\mathrm{loc}}(\Omega;\mathbb R^n)$ be a nonconstant sense-preserving mapping with dilatation…

Let $n\geq 2$, let $\Omega\subset\mathbb R^n$ be a domain, and let $F\colon\Omega\to\mathbb R^n$ be an orientation-preserving mapping in…

Let $M_\alpha$ denote the class of $\alpha$-convex functions, let $n=1/\alpha$ with $n\in\mathbb{N}$ and $n\geq 2$, and let the Schwarzian norm be…

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

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

Logarithmic coefficient bound. The inequality

Higher-dimensional openness and discreteness conjecture. In higher dimensions, every non-constant mapping $f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$ of finite distortion wi…

Conjectured order formula.

Surjectivity conjecture. The functional $\rho_{\mid\mathscr{B}_a}$ is onto $(0,1]$. Since the preceding theorem establishes only $\rho(\mathscr{B}_a)\subseteq(0,1]$, the conjecture…

Bhowmik–Maity's conjecture. The constant $A_p$ is finite for every $p\in(0,1)$.

Let $\mathcal{S}_e^{\ast}$ be the class of functions $f(z)=z+a_2z^2+a_3z^3+\cdots$ satisfying … and write … The coefficients $\gamma_n$ are the logarithmic coefficients of $f$. Log…

Let $\mathcal{S}$ denote the class of normalized univalent functions on the unit disk, and write … Here $\gamma_n$ are the logarithmic coefficients of $f$. Miliin's conjecture. For…

Cho et al.'s conjecture. The radius of convexity of the class $\mathcal{BS}(\alpha)$ is