143 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…

Milin's conjecture. For every $f\in\mathcal{S}$ and every integer $n\geq 2$,

Let $B(\tau)$ and $B_b(\tau)$ denote the universal integral means spectra of univalent and bounded univalent functions, respectively. Brennan conjecture. The equality … should hold…

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…

Pommerenke's conjecture. For every $f\in\mathcal{S}^*$ and every integer $n\geq 2$,

Mocanu's conjecture. Then the harmonic function $f=h+\overline{g}$ is univalent in $\mathbb{D}$. This conjecture concerns a sufficient condition for univalence of a sense-preservin…

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

Let $\mathcal{C}$ denote the class of convex univalent functions normalized by $g(0)=0$ and $g'(0)=1$, and let … where $\mathbb D_r=\{z:|z|<r\}$, $0<r\leq 1$. Yamashita's conjectur…

Bombieri's odd-even refinement. If

Let $f(z)=z+\sum_{j=2}^{\infty}a_jz^j$ be an odd, equivalently $2$-symmetric, schlicht function in the unit disc. Littlewood–Paley conjecture. All coefficients of $f$ are bounded i…

Univalence conjecture. The polynomial $F$ is univalent in the unit disk $\mathbb D$. Numerical evidence supports the assertion for all $N$, while it is established in the cases…

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…

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 $f\in\Sigma(0)$ be a univalent function with a quasiconformal extension, let $k(f)$ denote its Teichmüller norm, and let $\varkappa_p(f)$ denote its $p$-th truncated Grunsky no…

Robertson's conjecture. The coefficients of $g$ satisfy

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…

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

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 $\mathcal{A}$ be the class of analytic functions on the unit disk $\mathbb{D}:=\{z:|z|<1\}$ of the form … Let $\sigma$ be the class of functions $f\in\mathcal{A}$ such that bot…

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…

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 $\Sigma_0$ 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 critical set associated with $f$, and let … where $p(z)$ is a polynomial in $z$ of degree at least two. Let $\{R_j\}$ be a finite collection of bounded regions i…

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…