41 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 $f=h+\overline{g}\in\mathcal{S}_{\mathcal{H}}$, 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 $\Sigma$ 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…

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

Let $\mathcal{S}^*(\varphi)$ denote the Ma–Minda starlike class, and let $\mathcal{S}^*_L$ be the corresponding logarithmic starlike class. For…

Let $\alpha\geq1$ and let $f\in\mathcal{B}_1(\alpha)$ have the normalized expansion referred to as (A-05). Sharp coefficient bound conjecture. For every $n\geq2$, … This is one of…

Let $S$ denote the class of normalized univalent functions. For $0\leq\alpha\leq1$, suppose that $g\in S$ has the expansion $g(z)=z+b_2z^2+\cdots$ and satisfies … Let…

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 $u\in\mathcal S_{(0,\alpha)-H}^0$ 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…

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

Let $f$ be meromorphic in the unit disk $\mathbb{D}=\{z:|z|<1\}$, have a pole at $z=p$, and have Taylor expansion … Suppose that, for some $0<\lambda<1$, … and denote this class by…

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 $\mathcal{U}(\lambda)$ be the class under consideration, and let … be a function in this class. The parameter $\lambda$ is the one defining $\mathcal{U}(\lambda)$. Coefficient…

Coefficient-bound conjecture. For every $n\geq 2$, the sharp bound

Let $f(z)\in \mathscr{S}^*_{\wp}$. The class $\mathscr{S}^*_{\wp}$ consists of normalized analytic functions satisfying … where $\wp$ is the cardioid-domain mapping used to define…

Coefficient conjecture for $\mathcal{U}(\lambda)$. For every $n\geq 2$, the sharp inequality

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

Let $f(z)=zh(z)\overline{g(z)}\in\mathcal{S}_{Lh}$, where $h$ and $g$ are given by the coefficient expansions in the source. Thus, for $n\geq 1$, $a_n$ and $b_n$ denote the corresp…

For $0<p<1$, let $C(k,p)$ be the coefficient-estimate constants considered in the paper. Strict monotonicity conjecture. The sequence $C(k,p)$ is strictly increasing. This is prese…