Convexity constant of a domain and applications (Q504899)

The authors introduce the convexity constant \(K(D)\) of a domain \(D\subset\mathbb C\) by \[ K(D)=\inf_{a,b\in D} \sup_{\gamma\in\Gamma(a,b,D)}\frac{|a-b|}{l(\gamma)} \] where \(\Gamma(a,b,D)\) is the family of all rectifiable arcs \(\gamma\subset D\) with distinct endpoints \(a\) and \(b\), and \(l(\gamma)\) denotes the length of \(\gamma\). It is proved that a simply connected domain \(D\) is convex if and only if \(K(D)=1\). From the other side, \(K(D)=0\) for slit domains. Given two domains \(\Omega\subset D\subset\mathbb C\), denote \(D_{\Omega}=\{z\in D: z\notin\overline{\Omega}\}\) where \(\overline{\Omega}\) is the closure of \(\Omega\). The authors show that \(K(D_{B(z_0,r)})=\frac{2}{\pi}\) for convex domains \(D\) when \(\overline{B(z_0,r)}:=\{z:|z-z_0|\leq r\}\subset D\), \(r>0\), and \(K(D_{S(z_0,r)})=\frac{1}{2}\) when \[ \overline{S(z_0,r)}:=\Big\{z:|\text{Re}\,(z-z_0)|\leq\frac{r}{2},\;|\text{Im}\,(z-z_0)|\leq\frac{r}{2}\Big\} \subset D,\;\;r>0. \] Similarly, they compute \(K(U_{A(z_0,\alpha,\beta)})\) for the unit disk \(U=B(0,1)\) and the angular region \(A(z_0,\alpha,\beta)\) with the vertex at \(z_0\in U\) and of opening \(|\alpha-\beta|\). The convexity constant of a domain is applied to derive an extension of the Ozaki-Nunokawa-Krzyz univalence criterion for non-convex domains. Namely, the authors prove that if a domain \(D\subset U\) contains the origin and an analytic function \(f:D\to\mathbb C\), \(f(0)=0\), \(f'(0)=1\), \(f(z)/z\neq0\), satisfies the condition \[ \left|\left(\frac{1}{f(z)}-\frac{1}{z}\right)'\right|\leq K(D),\;\;z\in D, \] then \(f\) is univalent in \(D\), and this result is sharp.