On the maximum of the conformal radius in famylies of domains under some additional conditions

We solve the problems on the maximum of the conformal radius $R(D,1)$ in the family $\mathcal D(R_0)$ of all simply connected domains $D\subset\mathbb C$ containing the points 0 and 1 and having a fixed value of the conformal radius $R(D,0)=R_0$, and in the family $\mathcal D(R_0,\rho)$ of domains from $\mathcal D(R_0)$ with given hyperbolic distance $\rho=\rho_D(0,1)$ between 0 and 1. Analogs of the mentioned problems for doubly-connected domains with given conformal module are considered. Solution of the above problems is based on results of general character in the theory of problems of extremal decomposition and related module problems. Bibl. 7 titles.