13 problems

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

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

Logarithmic-coefficient conjecture. For every $f\in\mathcal{S}_{ch}^*$ and every $n\in\mathbb{N}$,

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…

Kumar–Gangania conjecture. If $f\in\mathcal{S}_{\wp}^{*}$, then the sharp bound is

Let $\mathcal{S}^{*}_{\wp}$ denote the class of cardioid starlike functions associated with $\psi(z)=1+ze^z$, and let $r_0$ denote the Bohr radius for this class. Bohr-radius conje…

Let $f\in\mathscr{S}^*_{\wp}$, where $\mathscr{S}^*_{\wp}$ is the class of normalized analytic functions satisfying … and let $H_3(1)$ denote the third Hankel determinant of the Ta…

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…

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…

Let $\mathcal{M}$ be the class of analytic functions $f$ in the unit disk $\mathbb{D}$ normalized by $f(0)=f'(0)-1=0$ and satisfying … A function is starlike in $\mathbb{D}$ if it…

Let ${\mathcal{F}}$ be the class of locally univalent functions $f$ in ${\mathcal A}$ satisfying … For $f(z)=z+\sum_{k=2}^{\infty}a_kz^k$, let $s_n(f)(z)=z+\sum_{k=2}^{n}a_kz^k$ be…

Conjecture A. If $f\in\mathcal{K}_{H}^{0}$, then $f$ is fully starlike of order $\alpha$ in $|z|<r_S(\alpha)$. The conjecture seeks the sharp improvement of the known fully starlik…

Starlikeness conjecture for $\mathcal M(0)$. Every function in $\mathcal M(0)$ is starlike in $\mathbb D$.