53 problems

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

Let $\Sigma_0$ be the class of univalent meromorphic functions $F$ in the exterior of the unit disk, normalized by $F(z)=z+O(z^{-1})$ as $z\to\infty$. Let $K=F(\mathbb{B}^c)^c$ con…

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 $\mathcal{S}$ be the class of normalized univalent functions, let $\overline{\mathbf{S}}$ be the indicated closed Schwarzian space, let $S_f$ be the Schwarzian derivative of…

Let $\mathcal{U}_R$ be the indicated class of univalent rational functions, let $\mathbb{T}$ be the unit circle, and let $S_R$ denote the Schwarzian derivative of $R$. Critical-poi…

Let $B_b(\tau)$ be the universal integral means spectrum for bounded univalent functions, let $S_f$ denote the Schwarzian derivative of an extremal function $f$, and let…

Let $B_b(\tau)$ be the universal integral means spectrum for bounded univalent functions, let $N_f$ denote the Pre-Schwarzian derivative of an extremal function $f$, and let…

Let $f\in\mathcal{S}$ be a univalent function with logarithmic coefficients $\gamma_n$, defined by the logarithmic expansion associated with $f$. Conjecture on consecutive logarith…

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…

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

Boundary extremal-function conjecture. For each $t\in\mathbb{R}$, $B_b(t)$ has at least one extremal function whose Pre-Schwarzian derivative lies in $\partial\mathcal{T}$.

Let $\{\mathfrak L_\alpha\}_{\alpha\in[0,1]}$ be the filtration of $\mathcal G_0$ defined in the paper. The family is called strict when its inclusions are strict as the parameter…

Boundary conjecture. The function $f$ belongs to $\partial\mathfrak H_\alpha$ whenever $\alpha\in(0,2/3]$. The source notes that several other boundary functions are known, while t…

Let $\mathfrak A=\{\mathfrak A_\alpha\}_{\alpha\le1}$ be the analytic filtration, and let $F_\alpha$ denote the function used in its definition. For each $\alpha\le1$, consider the…

Let $T$ and $N$ be positive integers, and let $F^{(0)}$ be the polynomial proposed as the extremizer for the $T$-symmetric univalent-polynomial optimization problem. A schlicht pol…

Let $T$ and $N$ be positive integers, and consider the optimization problem over real-coefficient univalent polynomials in $\mathbb D=\{z:|z|<1\}$ with $T$-fold symmetry and degree…

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

Let $\mathcal{SL}^{*}$ denote the class of lemniscate starlike functions, associated with $\psi(z)=\sqrt{1+z}$. The Bohr-radius conjecture. The Bohr radius of $\mathcal{SL}^{*}$ is…

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

Sharp-value conjecture. If $f\in\mathcal F$, 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…

Hadamard-product convexity conjecture. For any $f_1,f_2\in\Omega$, one has

Let $p_N$ be a polynomial of degree $N$ in the normalized univalent class $\mathcal S$, and define its Koebe radius by … The Koebe radius is the infimum of the boundary modulus amo…