UDC 517.54

MATHEMATICS

I. A. ALEKSANDROV, A. S. SOROKIN

ON THE EXTENSION OF THE VARIATIONAL METHOD OF G. M. GOLUZIN—P. P. KUFAREV TO MULTIPLY CONNECTED DOMAINS

(Presented by Academician M. A. Lavrent’ev, 9 XI 1966)

The aim of the present work is to extend the well-known variational theorem of G. M. Goluzin \((^1)\) to finitely connected domains. Earlier P. P. Kufarev, jointly with N. V. Genina (Semukhina) \((^2)\) (see also \((^5)\)), using a new integral representation of the first variation given by him \((^3)\), extended G. M. Goluzin’s theorem to families of univalent functions in an annulus. Important results in the same direction were obtained, on the basis of other considerations, by M. Schiffer \((^6)\) and S. A. Gel’fer \((^7)\). From the main variational theorem (Theorem 4) of our work there follows the above-mentioned theorem of P. P. Kufarev. The formula indicated in Theorem 1, which generalizes Schwarz’s formula to the case of finitely connected circular domains, is a development and strengthening of the results of V. A. Zmorovich \((^8)\) and G. Meshkovskii \((^9)\). Starting from this formula and using a number of considerations of P. P. Kufarev \((^3)\), we obtain theorems on differentiability with respect to the parameter \(t\) of families of functions \(F(w,t)\) and \(\Phi(z,t)\), regular in a finitely connected domain, a variational theorem, a generalized Löwner equation, and also a variational formula for a finitely connected domain of the type of the variational formula of M. A. Lavrent’ev for the disk.

Let the real numbers \(R_k>0\) and the complex numbers \(a_k\) \((a_0=a_n=0)\), \(k=0,\ldots,n\), be such that the inequalities \(|a_i|<R_0-R_i\), \(|a_i-a_j|>R_i+R_j\), \(i,j=1,\ldots,n\), are satisfied.

Theorem 1. Let \(f(z)\) be a function regular and single-valued inside the intersection of the disk \(|z|<R_0\) with the exteriors of the disks \(|z-a_k|\le R_k\), \(k=1,\ldots,n\), whose real part on the boundary components \(C_k:\ |\zeta-a_k|=R_k\), \(k=0,\ldots,n\), assumes the values \(f_k(\zeta)\). Then the formula holds

\[ f(z)=\sum_{k=0}^{n}\frac{\delta_k}{2\pi i}\int_{C_k} f_k(\zeta)H_k(\zeta,z)\,\frac{d\zeta}{\zeta} \]

\[ -\frac{1}{4\pi i}\int_{C_0} f_0(\zeta)\,\frac{d\zeta}{\zeta} -\frac{1}{4\pi i}\int_{C_n} f_n(\zeta)\,\frac{d\zeta}{\zeta}+iD, \]

where \(D\) is a real number, \(\delta_0=1,\ \delta_k=-1,\ k=1,\ldots,n,\)

\[ H_k(\zeta,z)= \frac{\zeta+z}{\zeta-z} +\sum_{\nu=1}^{\infty}\left[ \sum_{k_1}^{2\nu}{}' p(\zeta,z;k_1,k_2,\ldots,k_{2\nu}) +\sum_{k_1}^{2\nu-1} p(\zeta,z;k,k_1,\ldots,k_{2\nu-1}) \right], \]

and

\[ p(\zeta,z;k_1,k_2,\ldots,k_{2\nu})= \]

\[ =\left[ \frac{A_{k_{2\nu}}z}{R_{k_{2\nu}}^{2}t_{2\nu-1}+a_{k_{2\nu}}t_{2\nu}} +\frac{A_{h-k_{2\nu}}}{t_{2\nu}} \right] \frac{\zeta}{R_{k_{2\nu}}^{2}t_{2\nu-1}-(z-a_{k_{2\nu}})t_{2\nu}} \prod_{i=1}^{2\nu}R_{k_i}^{2}, \]

\[ A_k=1+\frac{1}{2}(\delta_k-\delta_{n-k}),\qquad \sum_{k_1}^{k_{2\nu}}{}' = \sum_{\substack{k_1=0\\ k_1\ne k}}^{n} \sum_{\substack{k_2=0\\ k_2\ne k_1}}^{n} \cdots \sum_{\substack{k_{2\nu}=0\\ k_{2\nu}\ne k_{2\nu-1}}}^{n}. \]

The functions \(t_p\) are determined by the recurrence relations

\[ t_p=-b_{p-1}t_{p-1}+R_{k_{p-1}}^{2}t_{p-2},\quad t_0=1,\quad t_1=\zeta-a_{k_1}, \]

where \(b_p=a_{k_{p+1}}-a_{k_p}\), if \(p\) is even, and \(b_p=\bar a_{k_{p+1}}-\bar a_{k_p}\), if \(p\) is odd.

The functions \(f_k(\zeta)\) satisfy the conditions

\[ \frac{1}{2\pi i}\int_{C_m'} f_m(\zeta)\frac{d\zeta}{\zeta-a_m} = \sum_{\substack{k=0\\ k\ne m}}^{n} \frac{\delta_k}{2\pi i} \int_{C_k'} f_k(\zeta)S_k(\zeta)\frac{d\zeta}{\zeta}, \qquad m=1,\ldots,n, \]

where

\[ S_k(\zeta)=T_k[\zeta,\bar L_m(\infty)]+ \frac{\delta_k-1}{2}\,T_k(\zeta,\infty), \]

\[ T_k(\zeta,z)=\frac{1}{2}H_k(\zeta,z) -\frac{\zeta L_k'(\zeta)}{2L_k(\zeta)}\, \bar H_k[L_k(\zeta),\bar z], \qquad L_k(\zeta)=\bar a_k+\frac{R_k^2}{\zeta-a_k}. \]

Theorem 2. Let a family of finitely \((n+1)\)-connected domains \(G(w,t)\), \(a\le t\le b\), be given in the \(w\)-plane, with the following properties:

1) the boundary components \(C_k(t)\), \(k=0,\ldots,n\), of the domain \(G(w,t)\) are simple closed Jordan curves \(w=\Omega_k(\lambda,t)\), where \(0\le \lambda\le 2\pi\), \(k=0,\ldots,n\), and none of the \(C_k(t)\), \(k=1,\ldots,n\), lies inside another, while all of them lie inside \(C_0(t)\).

2) the functions \(\Omega_k(\lambda,t)\), \(k=0,\ldots,n\), are differentiable in \(t\), uniformly with respect to \(\lambda\), at \(t=t_0\), \(t_0\in [a,b]\).

3) the curves \(C_k(t_0): w=\Omega_k(\lambda,t_0)\), \(k=0,1,\ldots,n\), are analytic.

Further, let \(z=F(w,t)\) be a function mapping conformally \(G(w,t)\) onto the intersection of the disk \(|z|<R_0(t)\) with the exteriors of the disks \(|z-a_k|\le R_k(t)\), \(k=1,\ldots,n\), where the functions \(R_k(t)\), \(k=0,\ldots,n\), are differentiable at the point \(t=t_0\); let there exist

\[ \left.\frac{\partial}{\partial t}[\arg F(w,t)]\right|_{t=t_0}, \qquad R_k(t_0)=R_k. \]

Then \(F(w,t)\) is differentiable in \(t\), uniformly with respect to \(w\) inside \(G(w,t_0)\), at \(t=t_0\), and for the derivative \(F_t'(w,t_0)\) the formula holds

\[ \frac{\partial F(w,t_0)}{\partial t} = -F(w,t_0)\left\{ \frac{1}{4\pi i}\int_{|\zeta|=R_0} L_0(\zeta)\frac{d\zeta}{\zeta} + \frac{1}{4\pi i}\int_{|\zeta|=R_n} L_n(\zeta)\frac{d\zeta}{\zeta} -iD \right. \]

\[ \left. -\sum_{k=0}^{n}\frac{\delta_k}{2\pi i} \int_{|\zeta-a_k|=R_k} L_k(\zeta)H_k[\zeta,F(w,t_0)]\frac{d\zeta}{\zeta} \right\}, \]

where

\[ L_k(\zeta)= -\left. \frac{\partial}{\partial t} \ln\bigl|F[\Omega_k(\lambda,t),t_0]/R_k(t)\bigr| \right|_{t=t_0}. \]

In this case the relations

\[ \frac{1}{2\pi i} \int_{|\zeta-a_m|=R_m} L_m(\zeta)\frac{d\zeta}{\zeta-a_m} = \sum_{\substack{k=0\\ k\ne m}}^{n} \frac{\delta_k}{2\pi i} \int_{|\zeta-a_k|=R_k} L_k(\zeta)S_k(\zeta)\frac{d\zeta}{\zeta}, \]

\[ m=1,\ldots,n. \tag{*} \]

Theorem 3. Under the conditions of Theorem 2, the function \(w=\Phi(z,t)\), inverse to \(F(w,t)\), is differentiable in \(t\), uniformly with respect to \(z\) inside the intersection of the disk \(|z|<R_0(t)\) with the exteriors of the disks \(|z-a_k|\le R_k(t)\), \(k=1,\ldots,n\), at \(t=t_0\), and

\[ \left. \frac{\partial \Phi(z,t)}{\partial t} \right|_{t=t_0} = z\frac{\partial\Phi(z,t_0)}{\partial z} \left\{ \frac{1}{4\pi i}\int_{|\zeta|=R_0} L_0(\zeta)\frac{d\zeta}{\zeta} + \frac{1}{4\pi i}\int_{|\zeta|=R_n} L_n(\zeta)\frac{d\zeta}{\zeta} \right. \]

\[ \left. -\sum_{k=0}^{n}\frac{\delta_k}{2\pi i} \int_{|\zeta-a_k|=R_k} L_k(\zeta)H_k(\zeta,z)\frac{d\zeta}{\zeta} -iD \right\}. \]

where

\[ L_k(\xi)=\frac{d}{dt}\ln R_k(t_0)-\operatorname{Re}\left[\frac{1}{\xi\Phi'_{\zeta}(\xi,t)}\frac{\partial}{\partial t}\Omega_k(\lambda,t)\right]_{t=t_0}. \]

In this case the relations \((*)\) are satisfied.

If in Theorems 2 and 3 one sets \(n=0\) and \(n=1\), then the well-known theorems of P. P. Kufarev \((^{2-4})\) are obtained.

Theorem 4. Let a function \(f(z)\) be holomorphic and univalent in the intersection of the disk \(|z|<R_0\) with the exteriors of the disks \(|z-a_k|\leq R_k,\ k=1,\ldots,n\). Suppose that the function \(f_k(z,t)=f(z)+tg_k(z)\) is holomorphic and univalent for each \(t\in[0,T]\) in the annulus \(R_k<|z-a_k|\leq R_k+\varepsilon_k\) for small \(\varepsilon_k>0,\ k=1,\ldots,n\), and maps it onto a domain with boundary continua \(C_{R_k}(t)\) and \(C_k(t)\). Further, let \(f_0(z,t)=f(z)+tg_0(z)\), for each \(t\in[0,T]\), be holomorphic and univalent in the annulus \(R_0-\varepsilon_0\leq |z|<R_0,\ \varepsilon_0>0\), and map this annulus onto a domain with boundary continua \(C_0(t)\) and \(C_{R_0}(t)\). We shall assume that each \(C_{R_k}(t)\) lies inside \(C_k(t)\), \(k=1,\ldots,n\), \(C_{R_0}(t)\)—outside \(C_0(t)\), none of the curves \(C_k(t)\), \(k=1,\ldots,n\), lies inside another, and, at the same time, all of them lie inside \(C_0(t)\). Let the functions \(R_k(t)\), \(k=0,\ldots,n\), be differentiable at the point \(t=0\), \(R_k(0)=R_k\). Then the function \(\Phi(z,t)\), conformally and univalently mapping the domain \(K(t)\), which is the intersection of the disk \(|z|<R_0(t)\) with the exteriors of the disks \(|z-a_k|\leq R_k(t)\), \(k=1,\ldots,n\), onto the \((n+1)\)-connected domain \(G(w,t)\), whose boundary consists of the continua \(C_{R_k}(t)\), \(k=0,\ldots,n\), is representable inside \(K(t)\) in the form

\[ \Phi(z,t)=f(z)+tzf'(z)P(z)+o(t), \]

where

\[ P(z)=\sum_{k=0}^{n}\frac{\delta_k}{2\pi i}\lim_{\rho_k\to R_k}\int_{|\zeta-a_k|=\rho_k}\operatorname{Re}B_k(\zeta)H_k(\zeta,z)\frac{d\zeta}{\zeta} \]

\[ -\frac{d}{dt}\ln\sqrt{R_0(0)R_n(0)}-iD-\frac{1}{4\pi i}\lim_{\rho_0\to R_0}\int_{|\zeta|=\rho_0}\operatorname{Re}B_0(\zeta)\frac{d\zeta}{\zeta} \]

\[ -\frac{1}{4\pi i}\lim_{\rho_n\to R_n}\int_{|\zeta|=\rho_n}\operatorname{Re}B_n(\zeta)\frac{d\zeta}{\zeta},\qquad B_k(\zeta)=\frac{g_k(\zeta)}{\zeta f'(\zeta)}. \]

In this case

\[ \sum_{\substack{k=0\\ k\ne m}}^{n}\beta_k\frac{d}{dt}\ln R_k(0)-\frac{d}{dt}\ln R_m(0) =\sum_{\substack{k=0\\ k\ne m}}^{n}\frac{\delta_k}{2\pi i}\lim_{\rho\to R_k}\int_{|\zeta-a_m|=\rho}\operatorname{Re}B_k(\zeta)S_k(\zeta)\frac{d\zeta}{\zeta} \]

\[ -\frac{1}{2\pi i}\lim_{\rho\to R_m+0}\int_{|\zeta-a_m|=\rho}\frac{\operatorname{Re}B_m(\zeta)\,d\zeta}{\zeta-a_m},\qquad m=1,\ldots,n, \]

\[ \beta_k=\frac{\delta_k}{2\pi i}\lim_{\rho\to R_k}\int_{|\zeta-a_k|=\rho}S_k(\zeta)\frac{d\zeta}{\zeta}. \]

If in Theorem 4 one sets \(n=1\), then the theorem of P. P. Kufarev and N. V. Genina (Semukhina) \((^{2,5})\) is obtained. If, then, \(R_1\) is allowed to tend to zero, we obtain the theorem of P. P. Kufarev \((^4)\).

Theorem 5. Let a family of finite \((n+1)\)-connected domains \(G(w,t)\), \(t\in[\tau_0,0]\), be such that:

1) \(w_0\in G(w,t)\);

2) \(G(w,t_1)\subset G(w,t_2)\) for \(t_1>t_2;\ t_1,t_2\in(\tau_0,0]\);

3) the boundary components \(C_k(t)\), \(k=0,\ldots,n\), of the domain \(G(w,t)\) are simple closed Jordan curves \(\Omega_k(\theta,t)\), uniformly with respect to \(\theta\), \(0\leq\theta\leq 2\pi\), differentiable with respect to \(t\) on \((\tau_0,0]\).

Then the function \(w=\Phi(z,t)\), \(\Phi(z_0,t)=w_0\), mapping the intersection of the disk \(|z|<R_0(t)\) with the exterior of the disks \(|z-a_k|\leqslant R_k(t)\), \(k=1,\ldots,n\), onto the domain \(G(w,t)\), satisfies at \((t_0,0)\) the equation

\[ \Phi_t(z,t)\big|_{t=t_0} = z\Phi_z(z,t_0) \left\{ \frac{1}{4\pi}\Psi_0(2\pi,t_0) + \frac{1}{4\pi}\Psi_n(2\pi,t_0) - iD - \sum_{k=0}^{n} \frac{\delta_k}{2\pi} \int_{0}^{2\pi} H_k[a_k+R_k(t)e^{i\theta},z]\,d\Psi_k(\theta,t_0) \right\}, \]

where

\[ \Psi_k(\theta,t_0) = - \lim_{\rho_k=1} \int_{0}^{\theta} \frac{\partial}{\partial t} \ln \left| \frac{ F\{\Phi[a_k+\rho_k R_k(t)e^{i\theta},t],t_0\} }{ R_k(t) } \right|_{t=t_0} \,d\theta, \]

\[ \Psi_m(2\pi,t_0) = \sum_{\substack{k=0\\ k\ne m}}^{n} \lim_{\rho_k\to 1} \delta_k \int_{0}^{2\pi} B_k[\theta,\rho_k R_k(t_0)]\,d\Psi_k(\theta,t_0), \qquad m=1,\ldots,n, \]

where

\[ B_k(\theta,\rho) = -\frac{\rho e^{i\theta}}{a_k+\rho e^{i\theta}}\,S_k(a_k+\rho e^{i\theta}), \qquad F(w,t)\equiv \Phi^{-1}(z,t). \]

Theorem 5 is a direct generalization of the Löwner-type equation to families of finitely connected domains, obtained earlier for a family of simply connected domains by P. P. Kufarev \((^3)\), and for doubly connected domains by I. A. Aleksandrov \((^{10})\).

Theorem 6. Let the \((n+1)\)-connected domain \(G\) have boundary components

\[ C_k:\quad w_k(\theta)=a_k+R_k[1+\sigma_k(\theta)]e^{i\theta}, \qquad 0\leq \theta\leq 2\pi,\quad \operatorname{Im}\sigma_k(\theta)=0, \]

\(k=0,\ldots,n\). We shall assume that \(|\sigma_k(\theta)|<\varepsilon\), where \(\varepsilon>0\) and is small, and that \(C_k\) is star-shaped with respect to \(a_k\). Let \(\Phi(z)\) map conformally and univalently the domain \(K\), formed by the intersection of \(|z|<R_0\) with the exterior of the disks \(|z-a_k|\leqslant R_k\), \(k=1,\ldots,n\), onto \(G\). Then, up to small terms of order \(o(\varepsilon)\) inside \(K\), the function \(\Phi(z)\) is represented in the form

\[ \Phi(z)\simeq z\left\{ 1+ \sum_{k=0}^{n} \frac{\delta_k}{2\pi} \int_{0}^{2\pi} \sigma_k(\theta)H_k(a_k+R_ke^{i\theta},z) \frac{d\theta}{1+a_kR_k^{-1}e^{-i\theta}} - \frac{1}{4\pi} \int_{0}^{2\pi} \sigma_0(\theta)\,d\theta - \frac{1}{4\pi} \int_{0}^{2\pi} \sigma_n(\theta)\,d\theta - \frac{d}{dt}\ln \sqrt{R_0(0)R_n(0)} \right\}. \]

Moreover,

\[ \sum_{\substack{k=0\\ k\ne m}}^{n} \beta_k\frac{d}{dt}\ln R_k(0) - \frac{d}{dt}\ln R_m(0) = \]

\[ = \sum_{\substack{k=0\\ k\ne m}}^{n} \frac{\delta_k}{2\pi} \int_{0}^{2\pi} \sigma_k(\theta)B_k(\theta,R_k)\,d\theta - \frac{1}{2\pi} \int_{0}^{2\pi} \sigma_m(\theta)\,d\theta, \qquad m=1,\ldots,n. \]

From Theorem 6, by simple means, one derives the results obtained earlier for \(n=0\) by M. A. Lavrent’ev \((^{11})\), for \(n=1\) by I. A. Aleksandrov \((^{10})\) and G. V. Siryk \((^{12})\).

Tomsk State University
named after V. V. Kuibyshev

Received
29 X 1966

CITED LITERATURE

\({}^1\) G. M. Goluzin, Mat. sbornik, 19 (61), 2, 203 (1946).
\({}^2\) P. P. Kufarev, DAN, 107, No. 4 (1956).
\({}^3\) P. P. Kufarev, Mat. sbornik, 13 (55), 1 (1943).
\({}^4\) P. P. Kufarev, DAN, 97, No. 3 (1954).
\({}^5\) N. V. Genina (Semukhina), DAN, 107, No. 4 (1956); Uch. zap. TGU, No. 44, 226 (1962).
\({}^6\) M. Schiffer, Am. J. Math., 65, 341 (1943).
\({}^7\) S. A. Gel’fer, DAN, 142, No. 3 (1962).
\({}^8\) V. A. Zmorovich, Dokl. AN USSR, 5, 489 (1958).
\({}^9\) H. Meschkowski, Math. Zs., 62, 161 (1955).
\({}^{10}\) I. A. Aleksandrov, Sibirsk. matem. zhurn., 4, No. 5, 961 (1963).
\({}^{11}\) M. A. Lavrent’ev, B. V. Shabat, Methods of the Theory of Functions of a Complex Variable, 1951.
\({}^{12}\) G. V. Siryk, Izv. vyssh. uchebn. zaved., matem., 5 (1960).