I. M. Milin

CLOSED ORTHONORMAL SYSTEMS OF ANALYTIC FUNCTIONS IN FINITELY CONNECTED DOMAINS

(Presented by Academician V. I. Smirnov on 20 IV 1964)

Notation. Let \(B\) be a finitely connected domain of the \(z\)-plane, containing the point at infinity, with boundary \(K\), consisting of \(m\) closed mutually exterior analytic Jordan curves \(K^{(1)}, K^{(2)}, \ldots, K^{(m)}\); \(\overline B = B \cup K\); \(G(z,\infty,B)\) is the Green function for the domain \(B\) with pole at \(z=\infty\); \(K_\rho\) \((1 \le \rho < \infty)\) is a level line of the Green function, i.e. \(K_\rho = E\bigl(G(z,\infty;B)=\ln \rho\bigr)\); \(B_\rho\) is the domain containing \(z=\infty\) and bounded by the level line \(K_\rho\); \(B_{r,R}\) \((1 \le r < R)\) is the annular set \(E\bigl(\ln r < G(z,\infty;B) < \ln R\bigr)\).

We consider the following classes of functions: \(\sigma(B)\) is the class of functions \(f(z)\), regular* in the domain \(B\), \(f(\infty)=0\), for which
\[ \iint\limits_B |f'(z)|^2\,d\sigma < \infty; \]
\(l^2(B)\) is the class of functions \(f'(z)\), where \(f(z)\in\sigma(B)\); \(\Sigma(B)\) is the class of meromorphic and single-valued functions \(F(z)\) in the domain \(B\), having a single pole at the point at infinity and a Laurent expansion in its neighborhood of the form \(F(z)=z+\alpha_0+\alpha_1 z^{-1}+\cdots\); \(\widetilde{\Sigma}(B)\) is the subclass of \(\Sigma(B)\) of functions \(F(z)\) mapping \(B\) onto domains without exterior points and with zero boundary area.

I. Laurent system of functions

1. Existence and uniqueness of the Laurent system

We shall call a Taylor system of functions for the domain \(B\) a system of functions \(\{\varphi_n(z)\}\) \((n=1,2,\ldots)\) satisfying the conditions: a) the functions \(\varphi_n(z)\) \((n=1,2,\ldots)\) are regular in the closed domain \(\overline B\) and have, in a neighborhood of \(z=\infty\), a Taylor expansion of the form
\[ \varphi_n(z)=\sum_{k=n}^{\infty} a_{nk} z^{-k}, \qquad a_{nn}>0 \quad (n=1,2,\ldots); \tag{1} \]
b) the derivatives \(\varphi'_n(z)\) \((n=1,2,\ldots)\) form an orthonormal system in the domain \(B\), i.e.
\[ \frac{1}{\pi}\iint\limits_B \varphi'_n(z)\,\overline{\varphi'_k(z)}\,d\sigma = \delta_{nk}\quad (n,k=1,2,\ldots), \]
complete in the class \(l^2(B)\).

We shall call the conjugate system of functions (for the Taylor system of the domain \(B\)) a system of functions \(\{\Phi_n(z)\}\) \((n=1,2,\ldots)\) satisfying the conditions: a) the functions \(\Phi_n(z)\) \((n=1,2,\ldots)\) are regular in the closed domain \(\overline B\) except for a pole of order \(n\) at the point \(z=\infty\), and their Laurent expansion in a neighborhood of \(z=\infty\) contains no constant term; b) on each boundary curve \(K^{(\nu)}\) \((\nu=1,2,\ldots,m)\) the relation
\[ \Phi_n(z)=\overline{\varphi_n(z)}+K_{n\nu} \quad (n=1,2,\ldots;\ \nu=1,2,\ldots,m), \tag{2} \]
holds, where \(K_{n\nu}\) is a constant.

Finally, by a Laurent system for the domain \(B\) we shall mean a system of pairs of functions
\[ \{\varphi_n(z),\Phi_n(z)\}\quad (n=1,2,\ldots). \]

* By functions regular (meromorphic) in \(\overline B\) we mean single-valued regular (meromorphic) functions.

We shall now set forth some of the results obtained that are used for the study of univalent functions.

Theorem 1. For the given domain \(B\), the Taylor and Laurent systems of functions exist and are unique.

The proof uses a well-known result of Grunsky \((^1)\) stating that for every polynomial \(P(z)\) there exists a unique function \(u(z)=P(z)+f(z)\), where \(f(z)\) is regular in \(\overline{B}\), \(f(\infty)=0\), which maps the boundary curves \(K^{(\nu)}\) \((\nu=1,2,\ldots,m)\) one-to-one onto segments of parallel straight lines of a prescribed inclination to the real axis. The existence of the Taylor system was proved by Bergman \((^2)\). Systems of functions close to the Taylor system were considered by Walsh \((^3)\). If the domain \(B\) is the exterior of the unit circle, then \(\varphi_n(z)=\dfrac{1}{\sqrt n}z^{-n}\), \(\Phi_n(z)=\dfrac{1}{\sqrt n}z^n\) \((n=1,2,\ldots)\).

2. Properties of the functions of the Laurent system.

a) For the functions of the Laurent system the following equalities hold
\[ \frac{i}{2\pi}\oint_K \varphi_k(z)\,d\overline{\varphi_n(z)}=\delta_{kn},\qquad \frac{i}{2\pi}\int_K \varphi_k(z)\,d\varphi_n(z)=0 \quad (n,k=1,2,\ldots), \tag{3} \]
\[ \frac{i}{2\pi}\oint_K \Phi_k(z)\,d\overline{\Phi_n(z)}=-\delta_{kn},\qquad \frac{i}{2\pi}\int_K \Phi_k(z)\,d\Phi_n(z)=0 \quad (n,k=1,2,\ldots). \tag{4} \]

b) For the functions of the Taylor system \(\{\varphi_n(z)\}\), for each \(z\) in the domain \(B\) we have:
\[ \lim_{n\to\infty}\left|\varphi_n^{(\nu)}(z)\right|^{1/n} =e^{-G(z,\infty,B)}=\rho^{-1},\qquad z\in B \quad (\nu=0,1,\ldots). \tag{5} \]
Moreover, for any \(\varepsilon>0\) the inequality holds
\[ \left|\varphi_n^{(\nu)}(z)\right|\le M_\nu(1+\varepsilon)^{n+\nu}\rho^{-(n+\nu)}, \qquad z\in \overline{B} \quad (n=1,2,\ldots;\ \nu=0,1,\ldots), \tag{6} \]
where \(M_\nu\) depends only on \(\varepsilon\) and \(\nu\).

c) For the functions of the conjugate system \(\{\Phi_n(z)\}\), for each finite \(z\) in the domain \(B\) we have:
\[ \lim_{n\to\infty}\left|\Phi_n^{(\nu)}(z)\right|^{1/n} =e^{G(z,\infty;B)}=\rho,\qquad z\in B_{1,\infty} \quad (\nu=1,2,\ldots). \tag{7} \]
Moreover, for any \(\varepsilon>0\) the inequality holds:
\[ \left|\Phi_n'(z)\right|\le M(1+\varepsilon)^{n-1}\rho^{\,n-1}, \qquad z\in B_{1,\infty}\cup K \quad (n=1,2,\ldots), \tag{8} \]
where \(M\) depends only on \(\varepsilon\).

3. Expansion in a series with respect to the Taylor system.

With the aid of the properties of the functions of the Laurent system listed above, the domain of uniform convergence of the series
\(\sum_{n=1}^{\infty}\lambda_n\varphi_n(z)\),
\(\sum_{n=1}^{\infty}\lambda_n\varphi_n'(z)\)
is found, and the following is established.

Theorem 2. Every function \(f(z)\), regular in the domain \(B_r\), \(1\le r<\infty\), with Taylor expansion about \(z=\infty\) of the form
\[ f(z)=\sum_{k=1}^{\infty} d_k z^{-k} \tag{9} \]
can be expanded in a series with respect to the Taylor system of functions
\[ f(z)=\sum_{n=1}^{\infty}\lambda_n\varphi_n(z), \tag{10} \]
converging absolutely and uniformly inside the domain \(B_r\). Such an expansion is unique, and the expansion coefficients \(\lambda_n\) \((n=1,2,\ldots)\) are determined by the formula
\[ \lambda_n=\frac{i}{2\pi}\oint_{K_\rho} f(z)\,d\Phi_n(z) \quad (n=1,2,\ldots),\qquad r<\rho<\infty, \tag{11} \]

and satisfy the inequality

\[ \overline{\lim_{n\to\infty}}\,|\lambda_n|^{1/n}\leq r. \tag{12} \]

The quantities \(\lambda_n\) depend linearly on \(d_1,d_2,\ldots,d_n\) \((n=1,2,\ldots)\). If the function \(f(z)\) is regular in the domain \(B\), then the area of the image \(B\) (finite or infinite) is computed by the formula

\[ \sigma(f)=\iint\limits_B |f'(z)|^2\,d\sigma =\pi\sum_{n=1}^{\infty}|\lambda_n|^2. \tag{13} \]

4. Expansion in a Laurent-type series. Let the function \(\psi(z)\) be regular in some boundary annular set \(B_{1,R}\) \((R>1)\). Put

\[ \sigma^*(\psi) = -\frac{i}{2}\oint\limits_K \psi(z)\overline{\psi'(z)}\,dz = \lim_{\rho\to 1} -\frac{i}{2}\oint\limits_{K_\rho}\psi(z)\overline{\psi'(z)}\,dz. \tag{14} \]

The above-mentioned properties of the Laurent system of functions make it possible to find the domain of uniform convergence of the Laurent-type series
\[ \sum_{n=1}^{\infty}\Lambda_n\Phi_n'(z)+ \sum_{n=1}^{\infty}\lambda_n\varphi_n'(z) \]
and to prove the following theorem.

Theorem 3. The derivative of any function \(\psi(z)\), regular in the annular set \(B_{r,R}\), \(1\leq r<R\), can be expanded in a Laurent-type series

\[ \psi'(z)= \sum_{n=1}^{\infty}\Lambda_n\Phi_n'(z) + \sum_{n=1}^{\infty}\lambda_n\varphi_n'(z), \tag{15} \]

converging absolutely and uniformly inside \(B_{r,R}\).

Moreover, such an expansion is unique, and the coefficients of the expansion \(\Lambda_n\) and \(\lambda_n\) \((n=1,2,\ldots)\) are determined by the formulas

\[ \Lambda_n = -\frac{i}{2\pi} \oint\limits_{K_\rho}\psi(z)\,d\overline{\varphi_n(z)} \quad (n=1,2,\ldots),\qquad r<\rho<R; \tag{16} \]

\[ \lambda_n = \frac{i}{2\pi} \oint\limits_{K_\rho}\psi(z)\,d\Phi_n(z) \quad (n=1,2,\ldots),\qquad r<\rho<R, \tag{17} \]

and satisfy the inequalities

\[ \overline{\lim_{n\to\infty}}\,|\Lambda_n|^{1/n}\leq \frac1R,\qquad \overline{\lim_{n\to\infty}}\,|\lambda_n|^{1/n}\leq r. \tag{18} \]

If the function \(\psi(z)\) is regular in the boundary annular set \(B_{1,R}\) \((R>1)\), then

\[ \sigma^*(\psi)= \pi\left( \sum_{n=1}^{\infty}|\Lambda_n|^2 - \sum_{n=1}^{\infty}|\lambda_n|^2 \right). \tag{19} \]

II. The system of functions \(\{C_n(z)\}\).

1. Orthonormality of the system \(\{C_n(z)\}\). It is known that for any prescribed \(t\in B\) \((t\ne\infty)\) and \(\theta\in[0,\pi)\) there exists a unique function \(j_\theta(z,t)\in \Sigma(B)\), \(j_\theta(t,t)=0\), which maps the domain \(B\) onto the whole plane with slits along arcs of logarithmic spirals of inclination \(\theta\) to the radius vector.

Fix an arbitrary \(t\in B\) and form the functions

\[ P(z,t)= \frac12\ln\frac{j_{\pi/2}(z,t)}{j_0(z,t)}, \qquad R(z,t)= \frac12\ln\frac{(z-t)^2}{j_{\pi/2}(z,t)j_0(z,t)}, \quad z,t\in B. \tag{20} \]

\(R(z,t)\) is a symmetric function of \(z\) and \(t\) in the domain \(B\), and, for fixed

for each \(t\in B\) is regular in \(\overline{B}\). \(P(z,t)\), for fixed \(t\in B\), is regular in \(\overline{B}\), and its expansion in a series with respect to the Taylor system has the form

\[ P(z,t)=\sum_{n=1}^{\infty} \overline{\varphi_n(t)}\,\varphi_n(z), \qquad z,t\in B . \tag{21} \]

Take an arbitrary function \(F(z)\in \Sigma(B)\) and an arbitrary \(t\in B\), and construct the function of \(z\)

\[ g(z,t)=\ln \frac{z-t}{F(z)-F(t)}-R(z,t), \qquad z,t\in B . \tag{22} \]

The function \(g(z,t)\) is regular in the domain \(B\) and, by Theorem 2, expands in a series with respect to the Taylor system of functions for the domain \(B\):

\[ g(z,t)=\sum_{n=1}^{\infty} C_n(t)\varphi_n(z), \qquad z,t\in B . \tag{23} \]

The expansion (23), for \(z\in B\), determines unique functions \(C_n(z)\) \((n=1,2,\ldots)\), which are regular in the domain \(B\). With the aid of a lemma generalizing to finitely connected domains a lemma previously proved by the author \((^4)\), the following theorems are established.

Theorem 4. For each system \(\{C_n(z)\}\) \((n=1,2,\ldots)\), generated by a function \(F(z)\in \widetilde{\Sigma}(B)\), the system of derivatives \(\{C'_n(z)\}\) is a complete orthonormal system of functions in the domain \(B\).

Theorem 5. For any system \(\{C_n(z)\}\), generated by a function \(F(z)\in \Sigma(B)\), the inequalities hold

\[ \sum_{n=1}^{\infty} |C_n(z)|^2 \leq P(z,z), \qquad \sum_{n=1}^{\infty} |C'_n(z)|^2 \leq K(z,\overline{z}), \qquad z\in B, \tag{24} \]

where \(K(z,\overline{t})\) is the Bergman kernel-function \((^2)\) for the domain \(B\) in the class \(l^2(B)\), and, more generally, the inequality

\[ \sum_{n=1}^{\infty} |C_n^{(\nu)}(z)|^2 \leq \sum_{n=1}^{\infty} |\varphi_n^{(\nu)}(z)|^2, \qquad z\in B\;(\nu=0,1,\ldots). \tag{25} \]

The equality sign occurs if and only if \(F(z)\in \widetilde{\Sigma}(B)\).

2. Extremal systems \(\{C_n(z)\}\). Using a known result of Schiffer \((^5)\) on Faber polynomials, one can prove the following theorem.

Theorem 6. For any sequence of complex numbers \(\{x_n\}\) \((n=1,2,\ldots)\), for which

\[ 0<\sum_{n=1}^{\infty}|x_n|^2<\infty, \]

there exists at least one function \(F(z)\) of the class \(\Sigma(B)\) such that the system of functions \(\{C_n(z)\}\) generated by it, for all \(z\in B\), satisfies the equality

\[ \sum_{n=1}^{\infty} x_n C_n(z) \equiv \sum_{n=1}^{\infty} \overline{x_n}\varphi_n(z), \qquad z\in B, \tag{26} \]

where \(\varphi_n(z)\) \((n=1,2,\ldots)\) are the functions of the Taylor system.

From Theorems 4, 5, and 6 follow both previously known and some new results for functions of the class \(\Sigma(B)\) concerning distortion theorems and conditions of univalence.

Received
25 III 1964

References Cited

\(^1\) H. Grunsky, Math. Zs., 45, 29 (1939).
\(^2\) S. Bergman, Math. Surveys, No. 5, N. Y., 1950.
\(^3\) J. L. Walsh, P. Davis, J. Anal. Math., 2, 1 (1952).
\(^4\) I. M. Milin, DAN, 154, No. 2 (1964).
\(^5\) M. Schiffer, Bull. Math. Soc., 54, No. 6, 503 (1948).