Some extremal problems for pairs of functions without common values

By the method of the moduli of families of curves one solves some extremal problems in the family of pairs $\{f_1,f_2\}$ of functions $f_1(z)=\alpha z+\dotsb$, $f_2(z)=\beta z^{-1}+\beta_0+\beta_1z+\dotsb$, with real coefficients, univalent, regular, resp. meromorphic in the circle $\Delta=\{|z|<1\}$ and mapping onto nonoverlapping domains. As a special case the solution of a problem, posed by V.M. Miklyukov in Sib. Mat. Zh., Vol. 18, 1977, No. 5, pp. 1111–1124, is obtained.