15 problems

Let $d>1$ be the dimension and let $r:=1+\frac{1}{p}$, where $p$ is the exponent in the relevant $L_p$-norm. For the Lipschitz function class ${\cal L}$ considered in the paper, th…

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…

Schneider's conjecture. For $f_1,\ldots,f_n\in\operatorname{Conv}(\Omega)$, the mixed Hessian measure satisfies

Goodman–Saff conjecture. The radius $\sqrt{2}-1$ is best possible for the property that $f_r\in CV(i)$ whenever $f\in CV(i)$.

Intermediate-regime tightness conjecture. The intermediate regime in Theorem 1 is tight. Furthermore, the corresponding worst-case function is bivariate.

The functional B-conjecture. The function $\upalpha:\mathbb R\to\mathbb R_+$ is log-concave.

Let $n\geq 2$. A convex function is a function whose restriction to every line segment in its domain is convex. Write $B^n(0,1)$ for the open unit ball in $\mathbb R^n$, and let…

For $\lambda\in(0,1]$, let … where $U_f(z):=(z/f(z))^2f'(z)-1$, and let $\mathcal{U}_0(\lambda)$ consist of those functions in $\mathcal{U}(\lambda)$ satisfying $f”(0)=0$. A functi…

Let $m\geq 2$ and let $n$ be a positive integer. Denote by $w_{m,n}$ the global Whitney constant in dimension $n$, and by $w_{m,n}^{(s)}$ the corresponding supremum restricted to c…

Let ${\mathcal K}(\alpha)$ denote the class of normalized analytic functions that are convex of order $\alpha$, and let $k_\alpha$ be characterized by … with $k_\alpha(0)=0$ and…

Let $\mathcal{C}^0_H$ be the class of functions in $\mathcal{S}^0_H$ whose image of the unit disk is convex, and let $\mathcal{C}$ be the analytic class of normalized univalent fun…

Baricz–Ponnusamy's conjecture. $\varphi_{\nu}$ is convex in $\mathbb{D}$ if and only if $\nu\geq-1.875$.

Self-polar maximizer conjecture. The maximizer of

Conjecture B. If $f\in\mathcal{K}_{H}^{0}$, then $f$ is fully convex of order $\alpha$ in $|z|<r_S$, where $r_C=r_C(\alpha)$ is the positive root in $(0,1)$ of $p(r,u_0)=0$. The co…

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…