Yu. E. Alenitsyn

CONFORMAL MAPPINGS OF A MULTIPLY CONNECTED DOMAIN ONTO MULTI-SHEETED SURFACES WITH RECTILINEAR SLITS

(Presented by Academician V. I. Smirnov on 22 VI 1964)

Here the extremal properties of certain mappings of a finitely connected domain onto multi-sheeted Riemann surfaces with rectilinear slits are established.

Let \(G\) be a bounded \(n\)-connected (\(n \ge 2\)) domain of the \(z\)-plane with boundary consisting of simple closed analytic curves \(C_\mu\), \(\mu=1,\ldots,n\); let \(\zeta_j\), \(j=1,\ldots,s\), be arbitrary distinct points of the domain \(G\); let \(\alpha_j\), \(\alpha_{0,j}\), \(j=1,\ldots,s\), be arbitrary coefficients with \(\sum_{j=1}^{s}\alpha_{0,j}=0\), but not all equal to zero;

\[ S(z;\zeta,\alpha)=\sum_{j=1}^{s}\left[\frac{\alpha_j}{z-\zeta_j}+\alpha_{0,j}\log(z-\zeta_j)\right] \]

is the singularity function; \(\theta\in(-\pi/2,\ \pi/2]\); \(n_1,n_2\) are arbitrary nonnegative integers with \(n_1+n_2\le n-1\).

The existence is proved of functions \(\Phi_{k,\theta}(z)=S(z;\zeta,\alpha)+F_{k,\theta}(z)\), \(k=1,2\), with functions \(F_{k,\theta}(z)\) regular in the closed domain \(G\), possessing the following properties: on each boundary component \(C_\mu\) of the domain \(G\), each branch of the function \(e^{-i\theta}\Phi_{k,\theta}(z)\), \(k=1,2\), for \(\mu=1,\ldots,n_1\), has constant imaginary part; for \(\mu=n_1+1,\ldots,n_1+n_2\), constant real part; for \(\mu=n_1+n_2+1,\ldots,n\), on each component \(C_\mu\) each branch of the function \(e^{-i\theta}\Phi_{1,\theta}(z)\) has constant imaginary part, while each branch of the function \(e^{-i\theta}\Phi_{2,\theta}(z)\) has constant real part. By these properties each of the functions \(\Phi_{k,\theta}(z)\), \(k=1,2\), is determined uniquely up to an additive constant.

It is also proved that the sequence of functions \(\Phi_{k,\theta}^{[\nu]}(z)\), \(\nu=1,2,\ldots\), analogously defined by domains \(G^{(\nu)}\), as \(\nu\to\infty\) approximating the domain \(G\) from within, converges uniformly inside this domain to the function \(\Phi_{k,\theta}(z)\), \(k=1,2\).

Consider the class \(\mathscr L_\theta(G;S;n_1,n_2)\) of all functions \(f(z)\) with the difference \(f(z)-S(z;\zeta,\alpha)\) regular in the domain \(G\) and on its boundary components \(C_\mu\), \(\mu=1,\ldots,n_1+n_2\), assigning to these boundary components, for \(\mu=1,\ldots,n_1\), segments of inclination \(\theta\) to the real axis, and for \(\mu=n_1+1,\ldots,n_1+n_2\), segments perpendicular to them (the precise meaning of this property was indicated above as applied to the functions \(\Phi_{k,\theta}(z)\), \(k=1,2\)). For \(n_1=n_2=0\) this class does not depend on \(\theta\) and is the class of all functions \(f(z)\) with the difference \(f(z)-S(z;\zeta,\alpha)\) regular in the domain \(G\).

Let \(\theta\) and the singularity function \(S(z;\zeta,\alpha)\) be given. Then on the class of all functions of the form

\[ f(z)=\sum_{j=1}^{s}\left[\frac{\gamma_j}{z-\zeta_j}+\gamma_{0,j}\log(z-\zeta_j)\right]+F(z), \]

where \(\{\gamma\}\) are arbitrary coefficients and \(F(z)\) is an arbitrary function regular in the domain \(G\), the functional

\[ I_\theta(f)=\operatorname{Re}\left\{e^{-2i\theta}\sum_{j=1}^{s}\left[\alpha_jF'(\zeta_j)-\alpha_{0,j}F(\zeta_j)\right]\right\} \]

is defined.

For a function \(f(z)\in\mathscr L_\theta(G;S;n_1,n_2)\), denote by \(\bar A(f)\) the exterior area of the function \(f(z)\) in the domain \(G\) (see \((^{1,2})\)).

Theorem 1. For any given \(\theta\) and any prescribed nonnegative \(\beta\) and \(\gamma\) with \(\beta+\gamma=1\), in the class \(\mathscr L_\theta(G;S;n_1,n_2)\) we have the sharp estimate:

\[ \bar A(f)+2\pi(\beta-\gamma)I_\theta(f)\le 2\pi\left[\beta^2 I_\theta(\Phi_{1,\theta})-\gamma^2 I_\theta(\Phi_{2,\theta})\right], \]

where the equality sign is attained for the function \(f=\beta\Phi_{1,\theta}+\gamma\Phi_{2,\theta}\), up to an additive constant, and only for it.

For \(\theta=0\) and \(S(z;\zeta,\alpha)=1/(z-\zeta)\) we obtain Jenkins’ theorem \({}^{3}\), proved by him under the additional assumption of regularity of the function \(f(z)\) on the entire boundary of the domain \(G\).

Put
\[ Q_{\theta}(z)=\frac12[\Phi_{1,\theta}(z)+\Phi_{2,\theta}(z)]. \]
Corollary. Among all functions of the class \(\mathcal L_{\theta}(G;S;n_1,n_2)\), the greatest exterior area in the domain \(G\) is given by the function \(Q_{\theta}(z)\), and, up to an additive constant, only by it.

For \(n_1=n_2=0\) the function \(Q_{\theta}(z)\) does not depend on \(\theta\), and in this case we obtain the already known result \({}^{1,2}\).

Put
\[ P_{\theta}(z)=\frac12[\Phi_{1,\theta}(z)-\Phi_{2,\theta}(z)] \]
and, for any function \(f(z)\) regular in the domain \(G\), denote by \(A(f)\) the area of the image of the domain \(G\) under the mapping \(w=f(z)\). We have
\[ A(P_{\theta})=A(Q_{\theta})=\pi I_{\theta}(P_{\theta}). \]
For any prescribed \(\theta\) and any prescribed function \(S(z;\zeta,\alpha)\), consider the class \(R_{\theta}(G;S;n_1,n_2)\) of all functions \(f(z)\) regular in the domain \(G\) and on its boundary components \(C_\mu\), \(\mu=1,\ldots,n_1+n_2\), which assign to the boundary components \(C_\mu\) of this domain, for \(\mu=1,\ldots,n_1\), segments of inclination \(\theta\) to the real axis, and for \(\mu=n_1+1,\ldots,n_1+n_2\), segments perpendicular to them, and which are normalized by the condition \(I_{\theta}(f)=1\). \(R_{\theta}(G;S;0,0)\) is the class of all functions regular in the domain \(G\) and normalized in the indicated manner.

Theorem 2. Among all functions of the class \(R_{\theta}(G;S;n_1,n_2)\), the least area of the image of the domain \(G\) is given by the function \(P_{\theta}(z)/I_{\theta}(P_{\theta})\), and, up to an additive constant, only by it.

Putting \(\theta=0\), \(S(z;\zeta,\alpha)=1/(z-\zeta)\), we obtain the solution of the problem of the least area of the image of the domain \(G\) in the class of all functions \(f(z)\) regular in this domain, assigning to its boundary components \(C_\mu\) horizontal (for \(\mu=1,\ldots,n_1\)) and vertical (for \(\mu=n_1+1,\ldots,n_1+n_2\)) segments and normalized by the condition \(\operatorname{Re} f'(\zeta)=1\), \(\zeta\in G\). Putting \(\theta=0\), \(S(z;\zeta,\alpha)=\log (z-\zeta_1)/(z-\zeta_2)\), \(\zeta_1,\zeta_2\in G\), we find the solution of this problem in the same class of regular functions, but normalized by the conditions: \(f(\zeta_1)=0\), \(\operatorname{Re} f(\zeta_2)=1\).

Let us also note the following consequence of Theorems 1 and 2.

Corollary. The product of the least area in the minimal problem considered for the class \(R_{\theta}(G;S;n_1,n_2)\) and the greatest exterior area in the maximal problem considered for the class \(\mathcal L_{\theta}(G;S;n_1,n_2)\) is equal to \(\pi^2\).

This generalizes, to the indicated classes of functions, the well-known \({}^{4}\) relation between the least area of the image of the domain \(G\) in the class of all functions regular in it with \(f'(\zeta)=1\), \(\zeta\in G\), and the greatest area of the complement to the image of this domain in the class of all univalent functions \(f(z)\) in it with regular difference \(f(z)-[1/(z-\zeta)]\).

For a doubly connected domain \(G\), all results generalize to the case of the singularity function
\[ S(z;\zeta,\alpha)=\sum_{j=1}^{s}\left[\sum_{k=1}^{p_j}\frac{\alpha_{k,j}}{(z-\zeta_j)^k}+\alpha_{0,j}\log(z-\zeta_j)\right] \]
and of the functional
\[ I_{\theta}(f)=\operatorname{Re}\left\{e^{-2i\theta}\sum_{j=1}^{s}\left[\sum_{k=1}^{p_j}\frac{\alpha_{k,j}}{(k-1)!}F^{(k)}(\zeta_j)-\alpha_{0,j}F(\zeta_j)\right]\right\}. \]

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
13 VI 1964

CITED LITERATURE

\({}^{1}\) Yu. E. Alenitsyn, DAN, 150, No. 4, 711 (1964).
\({}^{2}\) Yu. E. Alenitsyn, Izv. AN SSSR, Ser. Mat., 28, 609 (1964).
\({}^{3}\) J. A. Jenkins, Illinois J. Math., 7, No. 1, 104 (1952).
\({}^{4}\) M. Schiffer, Duke Math. J., 13, No. 4, 529 (1946).