Geodesic
Keldysh--Sedov formulas and differentiability with respect to the parameter of families of univalent functions in $n$-connected domains
Matematičeskie zametki, Tome 58 (1995) no. 6, pp. 878-889

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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. 
@article{MZM_1995_58_6_a7,
     author = {A. S. Sorokin},
     title = {Keldysh--Sedov formulas and differentiability with respect to the parameter of families of univalent functions in $n$-connected domains},
     journal = {Matemati\v{c}eskie zametki},
     pages = {878--889},
     publisher = {mathdoc},
     volume = {58},
     number = {6},
     year = {1995},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1995_58_6_a7/}
}
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A. S. Sorokin. Keldysh--Sedov formulas and differentiability with respect to the parameter of families of univalent functions in $n$-connected domains. Matematičeskie zametki, Tome 58 (1995) no. 6, pp. 878-889. http://geodesic.mathdoc.fr/item/MZM_1995_58_6_a7/