Keldysh–Sedov formulas and differentiability with respect to the parameter of families of univalent functions in $n$-connected domains

We introduce families of functions $F_j(w,t)$ mapping $(n+1)$-connected domains onto circular domains in the $z$-plane. Denote by $\Phi_j(z,t)$ the families of functions inverse to $F_j(w,t)$. Theorems 1-?4 treat differentiability properties of these families with respect to $t$ at a point $t=t_0$. We present formulas for the first derivative with respect to $t$. Corollaries of the theorems obtained are given. As a particular case, we deduce the theorem due to Kufarev for the disk and the theorem of Kufarev and Genina (Semukhina) for the annulus.