S. A. GELFER

ON THE EXTENSION OF THE GOLUZIN–SCHIFFER VARIATIONAL METHOD TO MULTIPLY CONNECTED DOMAINS

(Presented by Academician V. I. Smirnov on 31 VIII 1961)

1. Denote by (D) an (n)-connected domain of the (z)-plane, obtained by removing from the annulus (1<|z|<R) the interiors of (n-2) nondegenerate circles (L_k) ((k=2,\ldots,n-1;\ L_1: |z|<1,\ L_n: |z|<R)).

Let (F) be the class of functions (w=f(z)) possessing the following properties: 1) (f(z)) is regular and univalent in (D); 2) (f(z_0)=1), where (z_0) is a fixed point of the interval ((1,R)); 3) (f(z)\ne0,\ z\in D); 4) if the curve (l_1) is contained inside the curve (l_2), (l_1) and (l_2\subset D), then the curve (fl_1) is contained inside the curve (fl_2).

Let, further, (\Phi[f]) be a real functional, defined for all functions analytic in the domain (D), and such that for any function (g(z)) analytic in (D) the relation ((1)) holds

[
\Phi[f+\varepsilon g]=\Phi[f]+\operatorname{Re}{\varepsilon\Psi[f,g]}=O(\varepsilon^2),
\tag{1}
]

where (\Psi[f,g]) is a complex functional, linear with respect to (g(z)). Suppose that (\Phi[f]) has a finite upper bound in the class (F).

We pose the problem: to determine the functions of the class (F) for which (\Phi[f]) assumes its greatest value.

1. The existence of extremal functions follows from the normality of the family (F) in the domain (D).

We shall agree to denote by (D(L)) such a simply connected domain which contains (D) and has as its exact boundary the closed set (L) contained in the boundary of the domain (D). The following theorem is known ((^2)):

Let in a domain (D) of the (z)-plane there be given a univalent and single-valued conformal mapping (w=f(z)) of the domain (D) onto the (w)-plane. For any finite decomposition of the boundary (\Gamma=L_1\cup L_2\cup\cdots\cup L_n) of the domain (D) into nonempty pairwise disjoint closed sets there is a decomposition of (f) into a superposition

[
f(z)=f_n f_{n-1}\cdots f_1(z),\qquad z\in D,
\tag{2}
]

where the mappings (f_1,f_2,\ldots,f_n) are one-to-one and conformal respectively in the domains (D(L_1),\ f_1D(f_1L_2),\ldots,\ f_{n-1}\cdots f_1D(f_{n-1}\cdots f_1L_n)); such a decomposition is unique up to intermediate linear-fractional transformations.

1. We shall represent the extremal function (f(z)) of our problem by formula (2). We choose the component (t_1=f_1(z)) so that the conditions (f_1(\infty)=\infty,\ f_1(z)\ne0,\ f_1(z_0)=t_1^0) are satisfied, where (t_1^0) is determined from the relation (f_n f_{n-1}\cdots f_2(t_1^0)=1).

Using the variational formula of G. M. Goluzin (((^3),) formula (2), p. 125), we obtain a variation (\widetilde f_1(z)) for (f_1(z)); then, after a suitable normalization, the function (f_n f_{n-1}\cdots f_2\widetilde f_1(z)) will be a varied func-

functions in the class (F). The variational formulas have the form

[
\widetilde f
=
f+h\frac{f'}{f'_1}
\frac{f_1(f_1^0-f_1)}
{(f_1-t_1^)(f_1^0-t_1^)}
+O(\lambda^2),
\tag{3}
]

where (t_1^*) is an exterior point with respect to the domain (f_1D(f_1L_1));

[
\begin{aligned}
\widetilde f
&= f+\frac{f'}{f'_1}\Biggl[
-h\,\frac{f_1^{2}(f_1-f_1^0)}
{(f_1^-f_1)(f_1^-f_1^0)}
+h\,\frac{f_1^{3}}{z^{2}f_1^{'2}}
\left(
\frac{z^2 f'_1}{z^2-z}
-\frac{f_1}{f_1^0}\frac{z_0^2 f_1^{0'}}
{z^ - z_0}
\right)\
&\qquad\qquad
-\left(\frac{f_1^{3}}{z^{2}f_1^{'2}}\right)
\left(
\frac{z f'_1}{z^z-1}
-\frac{f_1}{f_1^0}\frac{z_0 f_1^{0'}}
{z^z_0-1}
\right)
\Biggr]+O(\lambda^2),
\end{aligned}
\tag{4}
]

where (\widetilde f=\widetilde f(z)), (f=f(z)), (f'=f'(z)), (f_1=f_1(z)), (f'_1=f'_1(z)), (f_1^=f_1(z^)), (f_1^{'}=f'_1(z^)), (f_1^0=f_1(z_0)), (f_1^{0'}=f'_1(z_0)), (z^*\in D), (h=\lambda e^{i\alpha}).

We shall have

[
\Phi[\widetilde f]\leqslant \Phi[f].
]

Hence, taking into account the arbitrariness of (\arg h), in the usual way we obtain for (f_1^*) the differential equation

[
\Psi\left[
f,\frac{f'}{f'_1}\frac{f_1(f_1-f_1^0)}{f_1^-f_1}
\right]
\frac{z^ f_1^{'2}}{f_1^(f_1^-f_1^0)}
=
-\Psi\left[
f,\frac{z_0^2 f_1^{0'} f_1 f'}{f'_1 f_1^0(z^-z_0)}
\right]
+
]

[
+\Psi\left[f,\frac{z^2 f'}{z^-z}\right]
+\Psi\left[
f,\frac{z_0 f_1^{0'} f_1 f'}{f'_1 f_1^0(z^z_0-1)}
\right]
-\Psi\left[f,\frac{z f'}{z^*z-1}\right].
\tag{5}
]

Here (f_1^*) denotes the component (f_1) of the extremal function in the superposition (2).

Consider the auxiliary function (\widetilde f_\theta(z)\in F):

[
\widetilde f_\theta(z)
=
f_n f_{n-1}\ldots f_2
\left(
f_1(z_0)\frac{f_1(e^{i\theta}z)}{f_1(e^{i\theta}z_0)}
\right)
=
f(z)+i\theta\Omega_1+O(\theta^2),
]

where

[
\Omega_1=
\frac{f}{f'_1}
\frac{z f'_1 f_1^0-z_0 f_1^{0'} f_1}{f_1^0},
]

(\theta) is arbitrary and real. From the condition (\Phi[\widetilde f_\theta]\leqslant \Phi[f]) we find that (\operatorname{Im}\Omega_1=0). Hence it is easy to see that the right-hand side of equation (5) is real for (|z^*|=1). Suppose further that

[
\Psi\left[
f,\frac{f'}{f'k f'}\ldots f'_1
\frac{f_k(f_k-f_k^0)}{f_k^*-f_k}
\right]
]

is a meromorphic function of (f_k^*) for all (k=1,2,\ldots,n).

By the principle of symmetry we conclude that the left-hand side of (5) ((k=1)) can be analytically continued into (|z|\leqslant 1). This shows that (f_1L_1) consists of analytic arcs.

Using formula (3), we shall prove that the domain (f_1D(f_1L_1)) has no exterior points and, consequently, is the whole (f_1)-plane with cuts along analytic arcs.

1. Let the function (\varphi_1(\zeta)), (\varphi_1(\infty)=\infty), map the domain (|\zeta|>1) onto the domain (f_1D(f_1L_2)). The function (\varphi(\zeta)) is analytic on (|\zeta|=1), since it maps the circle (|\zeta|=1) onto the analytic arc (f_1L_2) ((4), p. 164). We choose the component (t_2=f_2(f_1)) so that the conditions (f_2(\infty)=\infty), (f_2(f_1^0)\neq 0), (f_2(f_1^0=t_2^0)),

where (t_2^0) is determined from the relation (f_n f_{n-1}\cdots f_3(t_2^0)=1). We vary the function (f_2(\varphi(\zeta))), (|\zeta|>1), (f_1=\varphi_1(\zeta)); then, after a suitable normalization, the function (f_n f_{n-1}\cdots f_3 \widetilde f_2(f_1)) will be a varied function in the class (F). Carrying out the computations, as above, we obtain a differential equation for the component (f_2^*) of the extremal function (f):

[
\Psi\left[
f,\frac{f'}{f_2'f_1'}\,\frac{f_2(f_2-f_2^0)}{f_2^-f_2}
\right]\,
\frac{\zeta^{2}\varphi_1'{}^2(\zeta^)f_2^{\,\prime\,2}}
{f_2^(f_2^-f_2^0)}
=
]

[

\Psi\left[
f,\frac{\zeta^2\varphi_1'(\zeta)f'}{f_1'(\zeta^-\zeta)}
\right]
-
\Psi\left[
f,\frac{\zeta_0^2 f_2^0\varphi_1'(\zeta_0)f_2 f'}{f_2'f_1'f_2^0(\zeta^-\zeta_0)}
\right]
+
]

[
+
\overline{
\Psi\left[
f,\frac{\zeta_0\varphi_1'(\zeta_0)f_2^{0\,\prime}f_2 f'}{f_2'f_1'f_2^0(\zeta^\zeta_0-1)}
\right]}
\Psi
-
\overline{
\Psi\left[
f,\frac{\zeta\varphi_1'(\zeta)f'}{f_1'(\zeta^\zeta-1)}
\right]},
\tag{6}
]

where we have put
[
f_2=f_2(f_1),\quad
f_2^0=f_2(f_1^0),\quad
f_2^=f_2(f_1^),\quad
f_2'=f_2'(f_1),\quad
f_2^{0\,\prime}=f_2'(f_1^0),
]
[
f_2^{\,\prime}=f_2'(f_1^),\quad
\zeta=\varphi_1^{-1}(f_1),\quad
\zeta_0=\varphi_1^{-1}(f_1^0),\quad
\zeta^=\varphi_1^{-1}(f_1^).
]

We define the auxiliary variation by the formula

[
\widetilde f_\theta(z)
=
f_n f_{n-1}\cdots f_3
\left[
f_2(\varphi_1(\zeta_0))\,
\frac{f_2(\varphi_1(e^{i\theta}\zeta))}
{f_2(\varphi_1(e^{i\theta}\zeta_0))}
\right]
=
f(z)+i\theta\Omega_2+O(\theta^2),
]

where

[
\Omega_2=
\frac{f'}{f_2'f_1'}\,
\frac{
f_2^0 f_2\zeta\varphi_1'(\zeta)-f_2^{0\,\prime}f_2\zeta_0\varphi_1'(\zeta_0)
}
{f_2^0},
]
(\theta) is real and arbitrary.

From the condition (\Phi[\widetilde f_\theta]\leq \Phi[f]) we find that (\operatorname{Im}\Omega_2=0). Hence it is easy to see that the right-hand side of equation (6) is real for (|\zeta^*|=1). By the symmetry principle we conclude that the left-hand side of (6) can be analytically continued to (|\zeta|\leq 1); consequently, (f_2 f_1 L_2) consists of analytic arcs. The domain (f_2 f_1D\,(f_2 f_1 L_2)) has no exterior points and is the whole (f_2)-plane with slits along analytic arcs.

1. Let the function (\varphi_k(\zeta)), (\varphi_k(\infty)=\infty), (k=1,\ldots,n-1), map the domain (|\zeta|>1) onto the domain (f_k f_{k-1}\cdots f_1D\,(f_k\cdots f_1L_{k+1})). The functions (\varphi_k(\zeta)) ((k=1,\ldots,n-1)) are analytic on (|\zeta|=1), since they map the circle (|\zeta|=1) onto the analytic arcs (f_k\cdots f_1L_{k+1}). We choose the component (t_{k+1}=f_{k+1}(f_k)) so that the conditions (f_{k+1}(\infty)=\infty) for (k=1,\ldots,n-2) and (f_n(\infty)=0); (f_{k+1}(f_k)\ne0) ((k=1,\ldots,n-2)); (f_{k+1}(f_k^0)=t_{k+1}^0), where (t_{k+1}^0) is determined from the relation (f_n\cdots f_{k+2}(t_{k+1}^0)=1), are satisfied. For the component (f_{k+1}^*) of the extremal function we obtain a differential equation analogous to (6), from which we conclude that (f_{k+1}\cdots f_1L_{k+1}) ((k=1,\ldots,n-1)) consists of analytic arcs. The extremal function (f) maps the domain (D) onto the whole (w)-plane with (n) slits along piecewise-analytic arcs.

2. As an example, we consider the well-known problem of maximizing (|f'(z_0)|) in the class (F) for (n=2). Without loss of generality one may assume (f'(z_0)>0), and then the problem reduces to determining the maximum of (\operatorname{Re} f'(z_0)). In this case, for the extremal function one obtains the following explicit expression:

[
w=f(z)=
\frac{\operatorname{cn}u_0+\operatorname{dn}u_0\,\operatorname{cn}u-\operatorname{dn}u}
{\operatorname{cn}u_0-\operatorname{dn}u_0\,\operatorname{cn}u+\operatorname{dn}u},
\tag{7}
]

where
[
u=\frac{2K}{\ln R}\ln z,\quad
u_0=\frac{2K}{\ln R}\ln z_0,\quad
K=\int_0^1\frac{dx}{\sqrt{(1-x^2)(1-k^2x^2)}},\quad
\frac{2\pi}{\ln R}=\frac{K'}{K},
]
[
K'=\int_0^1\frac{dx}{\sqrt{(1-x^2)(1-k'^2x^2)}},\quad
k^2+k'^2=1,
]
(\operatorname{sn}u,\operatorname{cn}u,\operatorname{dn}u) are elliptic func-

Jacobi functions with modulus (k). The extremal function (f(z)) maps the annulus (1<|z|