Keldysh-Sedov formulas and differentiability with respect to the parameter of families of univalent functions in \(n\)-connected domains (Q1922321)

The author considers the families of functions \(F_j(w,t)\), \(j= 1,2\), mapping \((n + 1)\)-connected domains onto circular domains in the \(z\)-plane. The author denoting by \(\Phi_j (z,t)\), \(j=1,2\), the families of functions inverse to \(F_j (w,t)\), \(j=1.2\), proved four theorems which describe differential properties with respect to \(t\) of these families at a point \(t= t_0\). Among other things formulas for the first derivative over \(t\) are deduced. In particular, he obtained as a special case P. P. Kufarev's theorem for the disk [Dokl. Acad. Nauk SSSR, n. Ser. 97, 391-393 (1954; Zbl 0056.30001)] and the theorem of \textit{P. P. Kufarev} and \textit{N. V. Semukhina} [Dokl. Akad. Nauk SSSR 107, 505-507 (1956; Zbl 0073.06705)].