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

A. S. Sorokin

Siberian Metallurgical Institute

Abstract: 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.

Received: 28.09.1993

 English version: Mathematical Notes, 1995, 58:6, 1306–1314

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