17 problems

Hankel determinant conjecture. For every integer $n\ge3$,

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 $\mathcal{S}_{H}$ 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…

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

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

Let $0<\lambda\leq 1$, 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…

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…

Let $0<p<1$, and let $f$ be an extremal for $C(k,p)$, where $C(k,p)$ denotes the coefficient-estimate constant studied in the paper. Nonvanishing conjecture. Any such extremal $f$…

Liu–et al.'s coefficient conjectures. For every integer $n\geq 1$, the coefficients satisfy

Let $q$ satisfy $0<q<1$, let $0\leq\alpha<1$, and let $f\in\mathcal{S}_q^*(\alpha)$ have the normalized expansion … Let $\mu$ be any complex number. Fekete–Szegő conjecture. … Equa…

Let $\mathbb B_n$ be the unit ball in $\mathbb C^n$, and let $S^0(\mathbb B_n)$ be the normalized Loewner class. For $f\in S^0(\mathbb B_n)$, $D^kf(0)$ denotes its $k$th derivative…

Clunie–Sheil-Small coefficient conjecture. For every $n\geq2$,