On properties of the conformal radius of a domain

We consider the function $\rho(z)=\mathscr R(D,z)$, where $\mathscr R(D,z)$ is the conformal radius of a simply connected domain $D$ at a point $z\in D$. We study relations between the values of the function $\rho(z)$ at various points of the domain $D$. In Sec. 1, we establish exact inequalities relating the values of the function $\rho(z)$ in an arbitrary simply connected domain $D\subset\overline{\mathbb C}$ with the position of the conformal center and with the maximal conformal radius of the domain $D$. The same values are related to the values of $\rho(z)$ at another two points of the domain $D$. In Sec. 2, similar results are established for convex domains. This work supplements some recent results of Emel'yanov and Kovalev.