MATHEMATICS

Yu. E. ALENITSYN

CONFORMAL MAPPINGS OF A MULTIPLY CONNECTED DOMAIN ONTO MULTISHEETED CANONICAL SURFACES

(Presented by Academician V. I. Smirnov, 9 I 1963)

§ 1. Let \(G\) be a bounded \(m\)-connected domain of the \(z\)-plane with boundary \(C\), consisting of simple closed analytic curves \(C_1,\ldots,C_m\); \(\xi_1,\ldots,\xi_s\), \(s \geqslant 1\), be arbitrary mutually distinct points of the domain \(G\);

\[ \alpha_{0,j}, \alpha_{1,j},\ldots,\alpha_{p_j,j}, \quad j=1,\ldots,s, \]

be arbitrary coefficients with \(\sum_{j=0}^{s}\alpha_{0,j}=0\), but not all equal to zero (an admissible system of coefficients);

\[ S(z;\xi,\alpha)= \sum_{j=1}^{s}\left[\sum_{k=1}^{p_j}\frac{\alpha_{k,j}}{(z-\xi_j)^k}+\alpha_{0,j}\log(z-\xi_j)\right]. \]

Theorem 1. For any prescribed angle \(\theta\), \(-\pi/2<\theta\leqslant \pi/2\), and any prescribed function \(S(z;\xi,\alpha)\), there exists a function, unique up to an additive constant,

\[ \Phi_\theta(z)=S(z;\xi,\alpha)+F_\theta(z), \]

with \(F_\theta(z)\) regular* in the closed domain \(G\), which maps each boundary component of the domain \(G\) onto segments of certain straight lines making angle \(\theta\) with the real axis.

Corollary. Let \(\xi_1,\ldots,\xi_s\), \(s\geqslant 2\), be arbitrary prescribed mutually distinct points of the domain \(G\); let \(m_1,\ldots,m_s\) be arbitrary prescribed positive integers such that

\[ \sum_{j=1}^{k} m_j=\sum_{j'=k+1}^{s} m_{j'}; \]

let \(\theta\) be any prescribed angle, \(-\pi/2<\theta\leqslant \pi/2\). Then there exists a function \(\Psi_\theta(z)\), regular in the domain \(G\) except for poles at the points \(z=\xi_{j'}\), respectively of orders \(m_{j'}\), \(j'=k+1,\ldots,s\), with an expansion in a neighborhood of \(z=\xi_s\) of the form

\[ \Psi_\theta(z)=\frac{1}{(z-\xi_s)^{m_s}}+\cdots, \]

having in this domain zeros only at the points \(\xi_j\), respectively of multiplicities \(m_j\), \(j=1,\ldots,k\), and mapping each boundary component of it onto an arc of a certain logarithmic spiral making angle \(\theta\) with the rays issuing from the origin; such a function is unique.

We extend the notion of the exterior area of a meromorphic function \((^1)\) to functions of the form \(f(z)=S(z;\xi,\alpha)+g(z)\), where the function \(g(z)\) is regular in the domain \(G\). Namely, by the exterior area \(\overline{A}(f)\) of the function \(f(z)\) in the domain \(G\) we shall mean the limit (finite or infinite)

\[ \overline{A}(f)= -\lim_{\nu\to\infty}\frac{1}{2i}\int_{C^{(\nu)}} f'(z)\overline{f(z)}\,dz, \]

where \(C^{(\nu)}\) are the boundaries of domains \(G^{(\nu)}\) which, as \(\nu\to\infty\), approximate the domain \(G\) from within.

We introduce for consideration the functions

\[ P(z)=\frac{1}{2}\,[\Phi_{\pi/2}(z)-\Phi_0(z)],\qquad Q(z)=\frac{1}{2}\,[\Phi_{\pi/2}(z)+\Phi_0(z)], \]

corresponding to the prescribed function \(S(z;\xi,\alpha)\).

* By regular and meromorphic functions we mean single-valued regular and meromorphic functions.

It is easy to see that \(A(P)=\overline{A}(Q)\), where \(A(P)\) is the area of the image of the domain \(G\) under the mapping \(w=P(z)\).

Theorem 2. In the class of all functions of the form \(f(z)=S(z;\zeta,a)+g(z)\), where \(S(z;\zeta,a)\) is a fixed function and \(g(z)\) is an arbitrary function regular in the domain \(G\), the largest outer area is attained only by the function \(Q(z)\).

We shall regard the points \(\zeta_1,\ldots,\zeta_s\) as fixed and consider all possible admissible systems of coefficients \(\alpha_{0,j},\alpha_{1,j},\ldots,\alpha_{p_j,j}\), \(j=1,\ldots,s\). To each such system of coefficients there corresponds a function \(P(z)\), which we now normalize by the condition \(P(\zeta_s)=0\) and denote by \(P(z;\zeta_s)\).

Lemma. For any prescribed system of constants \(\beta_j^{(0)},\beta_j^{(1)},\ldots,\beta_j^{(p_j)}\), \(j=1,\ldots,s\), \(\beta_s^{(0)}=0\), not all zero simultaneously, there exists a unique system of coefficients \(\alpha_{0,j},\alpha_{1,j},\ldots,\alpha_{p_j,j}\), \(j=1,\ldots,s\), for which the corresponding function \(P(z;\zeta_s)\) satisfies the conditions
\[ P^{(k)}(\zeta_j;\zeta_s)=\beta_j^{(k)},\quad k=0,1,\ldots,p_j,\quad j=1,\ldots,s. \]

We denote the function \(P(z;\zeta_s)\) satisfying the conditions of the lemma by \(P_*(z;\zeta_s)\).

Theorem 3. Let any mutually distinct points \(\zeta_1,\ldots,\zeta_s\) of the domain \(G\) and any constants \(\beta_j^{(0)},\beta_j^{(1)},\ldots,\beta_j^{(p_j)}\), \(j=1,\ldots,s\), \(\beta_s^{(0)}=0\), not all zero, be given. In the class of all functions \(g(z)\), regular in the domain \(G\) and satisfying the conditions \(g^{(k)}(\zeta_j)=\beta_j^{(k)}\), \(k=0,1,\ldots,p_j\), \(j=1,\ldots,s\), the least value of the area \(A(g)\) is attained only by the function \(P_*(z;\zeta_s)\).

It is clear that this theorem also immediately yields the solution of the problem on the minimum area in the class of all functions regular in a given finitely connected domain and having, at prescribed points of it, arbitrary prescribed initial segments of their Taylor expansions. This problem was previously solved for a multiply connected domain in the class of regular functions with fixed values at two prescribed points of the domain \((^2)\), and in the class of regular functions with fixed values of several first derivatives at a prescribed point of the domain \((^1)\).

§ 2. Denote by \(\mathfrak{L}_{\zeta,a}(G)\) the class of all functions having the form \(f(z)=S(z;\zeta,a)+g(z)\), with fixed function \(S(z;\zeta,a)\), arbitrary function \(g(z)\) regular in the domain \(G\), and satisfying the condition \(\overline{A}(f)\ge 0\). Examples of functions of this class are the functions \(\Phi_\theta(z)\) and \(Q(z)\). The class \(\mathfrak{L}_{\zeta,a}(G)\) turns out to be convex. Let some class \(\overline{\mathfrak{L}}_{\zeta,a}(G)\) be considered. Then, for any function of the form
\[ f(z)=\sum_{j=1}^{s}\left[\sum_{k=1}^{p_j}\frac{\gamma_{k,j}}{(z-\zeta_j)^k}+\gamma_{0,j}\log(z-\zeta_j)\right]+F(z), \]
where \(\{\gamma\}\) are arbitrary coefficients and \(F(z)\) is an arbitrary function regular in the domain \(G\), the functional
\[ I(f)=\sum_{j=1}^{s}\left[\alpha_{0,j}F(\zeta_j)-\sum_{k=1}^{p_j}\frac{\alpha_{k,j}}{(k-1)!}F^{(k)}(\zeta_j)\right] \]
is defined.

In particular, \(A(P)=\pi I(P)\).

Let \(v_\nu(z)\), \(\nu=1,2,\ldots\), be an orthonormal system of functions of the class \(\mathcal H(G)\) (see \((^1)\)), complete in this class. Setting \((f',v_\nu)=-\dfrac{1}{2i}\int_C f\overline{v_\nu}\,d\bar z\), we have the theorem:

Theorem 4. If the function \(f(z)\in \mathfrak L_{\zeta,\alpha}(G)\) and its regular part is regular in the closed domain \(G\), then we have the sharp estimate

\[ \sum_{\nu=4}^{\infty}\left|(f',v_\nu)\right|^2 \leqslant \pi I(P). \]

Theorem 5. For any fixed \(\theta\), \(-\pi/2<\theta\leqslant \pi/2\), in the class \(\overline{\mathfrak L}_{\zeta,\alpha}(G)\) the extremal problem
\[ \operatorname{Re}\{e^{-2i\theta}I(f)\}=\min \]
is solved only by the function \(\Phi_\theta(z)\).

Theorem 6. If the function \(f(z)\) ranges over the class \(\overline{\mathfrak L}_{\zeta,\alpha}(G)\), then the range of values of the functional \(I(f)\) is the disk
\[ |w-I(Q)|\leqslant I(P), \]
and to each point on the boundary of this disk there corresponds only the function \(\Phi_\theta(z)\) with a suitably determined \(\theta\).

Theorem 7. In the subclass of all functions \(f(z)\) from \(\overline{\mathfrak L}_{\zeta,\alpha}(G)\) with any fixed value of \(\operatorname{Re}\{I(f)\}\), the greatest exterior area in the domain \(G\) is attained only by the function \(Q(z)+\lambda P(z)\) of this subclass, with a unique \(\lambda\) from the segment \([-1,1]\).

§ 3. The results of the preceding paragraphs can be applied to the study of functions meromorphic and \(p\)-valent on the average in the domain \(G\). Denote by \(\overline{\Sigma}_p(\zeta_1,\ldots,\zeta_k;\zeta)\) the class of all functions \(f(z)\), regular in the domain \(G\) except for a \(p\)-fold pole at its point \(\zeta\), with expansion in its neighborhood of the form
\[ f(z)=\frac{1}{(z-\zeta)^p}+\ldots, \]
having zeros at the points of the domain \(\zeta_j\), \(j=1,\ldots,k\), respectively of multiplicities \(m_j\), with
\[ \sum_{j=1}^{k} m_j=p, \]
and \(p\)-valent on the average in the domain \(G\) in the following sense: for any \(r>0\), the area of that part of the image of the domain \(G\) under the mapping \(w=f(z)\) which lies over the disk \(|w|<r\) does not exceed \(p\pi r^2\). By \(\overline{\Sigma}^{\,*}_p(\zeta_1,\ldots,\zeta_k;\zeta)\) denote the subclass of those functions from \(\overline{\Sigma}_p(\zeta_1,\ldots,\zeta_k;\zeta)\) for which \(\arg f(z)\) has single-valued branches in neighborhoods, lying in the domain \(G\), of its boundary components.

Theorem 8. If the function \(f(z)\) ranges over the class \(\overline{\Sigma}^{\,*}_p(\zeta_1,\ldots,\zeta_k;\zeta)\), then the range of values of the functional
\[ \sum_{j=1}^{k} m_j \log \frac{f^{(m_j)}(\zeta_j)}{q^{(m_j)}(\zeta_j)} \]
is the disk
\[ |w|\leqslant \sum_{j=1}^{k} m_j P(\zeta_j;\zeta), \]
and to each point on the boundary of this disk there corresponds only the function \(\Psi_\theta(z)\) with suitably determined \(\theta\). Here \(\Psi_\theta(z)\) is the derived function from Theorem 1, determined by \(s=k+1\) points \(\zeta_j\), \(\zeta_s=\zeta\), and coefficients \(m_j\), \(m_s=-p\), \(j=1,\ldots,k\); \(P(z;\zeta)\) corresponds to the function
\[ S(z;\zeta,a)=\sum_{j=1}^{s} a_{0,j}\log(z-\zeta_j) \]
with \(a_{0,j}=m_j\), \(j=1,\ldots,k\), \(a_{0,s}=-p\); \(q(z)\) is that one of the single-valued branches in \(G\) of
\[ \sqrt{\Psi_{\pi/2}(z)\Psi_0(z)} \]
for which, in a neighborhood of \(z=\zeta\), we have
\[ q(z)=\frac{1}{(z-\zeta)^p}+\ldots; \]
at the points \(\zeta_j\) one takes the values of that branch of
\[ \log\frac{f(z)}{q(z)} \]
which tends to zero as \(z\to\zeta\).

In the case when the domain \(G\) is the unit disk, all the results obtained become completely explicit. In particular, we note the theorem:

Theorem 9. In the class of all functions
\[ f(z)=z^p\left(1+\frac{a_1}{z}+\ldots\right), \]
\(p\)-valent and regular in the domain \(|z|>1\), except for the pole at \(z=\infty\),

having at the fixed point \(z_0 \ne \infty\) of this domain an expansion of the form
\(f(z)=f(z_0)+\dfrac{f^{(p)}(z_0)}{p!}(z-z_0)^p+\cdots\), the range of values of the functional
\(\log \dfrac{f^{(p)}(z_0)}{p!}\) is the circle
\(|w|\le -p\log\left(1-\dfrac{1}{|z_0|^2}\right)\).

In the case \(p=1\) this theorem turns out to be true for any fixed point \(z_0\) of the domain \(|z|>1\), and we obtain a result known for single-valued functions \((^3)\).

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

Received
2 I 1963

References

\(^1\) Yu. E. Alenitsyn, DAN, 146, No. 2 (1962).
\(^2\) H. Grunsky, Schr. d. math. Sem. u. d. Inst. f. angew. Math. an d. Univ. Berlin, 1, 94 (1932).
\(^3\) M. Grötzsch, Ber. d. Sächs. Akad. d. Wiss., Leipzig, 83, 283 (1931).