MATHEMATICS

Yu. E. Alenitsyn

On Some Extremal Properties of Functions Multivalent in Multiply Connected Domains

(Presented by Academician V. I. Smirnov on 9 IV 1962)

Let \(G\) be a bounded finitely connected domain of the \(z\)-plane with boundary \(C\), consisting of simple closed analytic curves; let \(\zeta\) be any prescribed point of the domain \(G\); let \(\alpha_1,\ldots,\alpha_p\) be any prescribed constants, not all equal to zero; and let \(\theta\) be any prescribed angle, \(0 \leq \theta < \pi\). Setting

\[ Q_p\left(\frac{1}{z-\zeta}\right)=\sum_{k=1}^{p}\frac{\alpha_k}{(z-\zeta)^k}, \]

we denote by \(\varphi_{\theta,p}(z,\zeta)\) that uniquely determined function which is regular* in the domain \(G\), except for a pole at the point \(z=\zeta\), has in its neighborhood an expansion of the form

\[ \varphi_{\theta,p}(z,\zeta) = Q_p\left(\frac{1}{z-\zeta}\right) + \sum_{n=1}^{\infty} a_n (z-\zeta)^n \]

and assigns to each boundary component of the domain \(G\) a segment of some straight line of inclination \(\theta\) to the real axis. Let \(g(z)\) be any function regular in the domain \(G\), and let \(A(g)\) be the area (finite or infinite) of the image of the domain \(G\) under the mapping \(w=g(z)\) (multiply covered area is counted with multiplicity). For a function \(f(z)\), regular in the domain \(G\) except for a finite number of poles, we introduce the exterior area \(\overline{A}(f)\) of the function \(f(z)\) in the domain \(G\). Namely, considering a sequence of domains \(G_k\) approximating the domain \(G\) from within, we set

\[ \overline{A}(f)=\lim_{k\to\infty}\frac{1}{2}\int_{B^{(k)}} R^2\,d\Phi, \]

where \(f(z)=Re^{i\Phi}\), \(B^{(k)}\) is the boundary of the image \(W_k\) of the domain \(G_k\) under the mapping \(w=f(z)\), and the integration is carried out in the negative direction with respect to \(W_k\). If the function \(w=f(z)\) is regular on the boundary of the domain \(G\) and assumes the value \(w=\infty\) in \(G\) \(p\) times, then \(\overline{A}(f)\) gives the difference between the area of the \(p\)-sheeted disk

\[ |w|<\max_{z\in C}|f(z)| \]

and the area of that part of the image of the domain \(G\) which lies over it. Put

\[ M_p(z,\zeta)=\frac{1}{2}\left[\varphi_{0,p}(z,\zeta)-\varphi_{\pi/2,p}(z,\zeta)\right], \]

\[ N_p(z,\zeta)=\frac{1}{2}\left[\varphi_{0,p}(z,\zeta)+\varphi_{\pi/2,p}(z,\zeta)\right]. \]

It is easy to see that \(A(M_p)=\overline{A}(N_p)\).

Theorem 1. In the class of all functions \(f(z)\), regular in the domain \(G\) except for poles at its points \(z=\zeta_\nu\), \(\nu=1,\ldots,s\), and having

* In what follows, by functions regular and meromorphic in a domain we mean single-valued functions regular and meromorphic in it.

at these points the principal parts

\[ Q_{p_\nu}\left(\frac{1}{z-\zeta_\nu}\right) = \sum_{k=1}^{p_\nu}\frac{\alpha_{k,\nu}}{(z-\zeta_\nu)^k}, \]

the function

\[ \sum_{\nu=1}^{s} N_{p_\nu}(z,\zeta_\nu) \]

and, up to an additive constant, only it gives the greatest exterior area.

This theorem generalizes the theorem known in \((^1)\) for univalent functions. Consider the set of all functions \(M_p(z,\zeta)\) corresponding, for fixed \(\zeta\) and \(p\), to all possible systems of coefficients \(\alpha_1,\ldots,\alpha_p\).

Lemma. For any given system of constants \(\beta_1,\ldots,\beta_p\), not all simultaneously equal to zero, there exists a unique system of coefficients \(\alpha_1,\ldots,\alpha_p\) for which the corresponding function \(M_p(z,\zeta)\) satisfies the conditions:

\[ M_p^{(l)}(\zeta,\zeta)=\beta_l,\qquad l=1,\ldots,p. \]

Denoting the function \(M_p^{*}(z,\zeta)\) indicated in this lemma by \(M_p^{*}(z,\zeta)\), we have the theorem:

Theorem 2. Let arbitrary constants \(\beta_1,\ldots,\beta_p\), not all simultaneously equal to zero, be given. In the class of all functions \(g(z)\), regular in the domain \(G\) and satisfying the conditions: \(g^{(l)}(\zeta)=\beta_l,\ l=1,\ldots,p\), the least value of the area \(A(g)\) is given by the function \(M_p^{*}(z,\zeta)\) and, up to an additive constant, only by it.

This theorem generalizes the theorem known in \((^1)\) for the case \(p=1\).

Let now the function \(f(z)\) be regular in the domain \(G\) except for a pole of order \(p\) at the point \(z=\zeta\). We shall call the function \(f(z)\) weakly \(p\)-sheeted in area in the domain \(G\) if for it \(\overline{A}(f)\ge 0\). Denote by \(\overline{\mathfrak{M}}_{p,\alpha}(G)\) the class of all functions weakly \(p\)-sheeted in area in the domain \(G\) and having fixed coefficients \(\alpha_1,\ldots,\alpha_p\) of the principal part

\[ Q_p\left(\frac{1}{z-\zeta}\right), \]

by \(\overline{\mathfrak{M}}_{p,\alpha}(G)\). It can be shown that the class \(\overline{\mathfrak{M}}_{p,\alpha}(G)\) is convex. Let \(\mathfrak{H}(G)\) be the class of all functions \(h(z)\), regular in the domain \(G\) together with their integrals and satisfying the condition

\[ \iint_G |h(z)|^2\,dx\,dy < \infty,\qquad z=x+iy, \]

and let \(v_\nu(z)\), \(\nu=1,2,\ldots\), be a system of functions of the class \(\mathfrak{H}(G)\), orthonormalized by the conditions

\[ \iint_G v_\mu(z)\overline{v_\nu(z)}\,dx\,dy = \begin{cases} 0, & \mu\ne \nu,\\ 1, & \mu=\nu, \end{cases} \]

and complete in this class. Putting

\[ (f',v_\nu)=-\frac{1}{2i}\int_C f\,\overline{v_\nu}\,d\overline{z},\qquad \nu=1,2,\ldots, \]

we obtain a theorem of the type of the well-known area theorem of Bieberbach:

Theorem 3. If the function \(f(z)\) belongs to the class \(\overline{\mathfrak{M}}_{p,\alpha}(G)\) and is regular on the boundary \(C\) of the domain \(G\), then

\[ \sum_{\nu=1}^{\infty}|(f',v_\nu)|^2 \le \pi\sum_{k=1}^{p}\frac{\alpha_k}{(k-1)!}\,M_p^{(k)}(\zeta,\zeta). \]

The estimate is sharp.

This theorem generalizes, on the one hand, Meshkovskii’s theorem \((^2)\) for functions univalent in a multiply connected domain, and on the other hand the corresponding result of the author \((^3)\) for functions weakly \(p\)-sheeted in area in the unit disk.

Let some class \(\overline{\mathfrak{M}}_{p,\alpha}(G)\) be considered. Then for any function \(F(z)\) having, in a neighborhood of the point \(z=\zeta\), an expansion in a power

a series with regular part of the form \(\sum_{n=0}^{\infty} b_n(z-\zeta)^n\), the functional is defined by

\[ J(E)=\sum_{k=1}^{p} k\alpha_k b_k . \]

Theorem 4. For any fixed \(\theta\), \(0\leq \theta<\pi\), the extremal problem
\(\operatorname{Re}\{e^{-2i\theta}J(f)\}=\max\) in the class \(\mathfrak{M}_{p,\alpha}(G)\) is solved by the function \(\varphi_{\theta,p}(z,\zeta)\), and, up to an additive constant, by it alone.

This theorem generalizes the corresponding result of Grunsky \((^4)\) for \(p\)-valent functions.

Theorem 5. If the function \(f(z)\) ranges over the class \(\mathfrak{M}_{p,\alpha}(G)\), then the range of values of the functional \(J(f)\) is the disk
\(\left|w-J\!\left(\overline{N_p}\right)\right|\leq J(M_p)\), and to each point on the boundary of this disk there corresponds a function \(\varphi_{\theta,p}(z,\zeta)\) with suitable \(\theta\), and, up to an additive constant, only it.

This theorem generalizes the well-known theorem of Grötzsch \((^5)\) for univalent functions.

For any \(\lambda\in[-1,1]\), the function
\[ L_\lambda(z,\zeta)=N_p(z,\zeta)+\lambda M_p(z,\zeta) \]
belongs to the class \(\mathfrak{M}_{p,\alpha}(G)\), and as \(\lambda\) increases from \(-1\) to \(1\), \(\operatorname{Re}\{J(L_\lambda)\}\), increasing, runs through the set of all values \(\operatorname{Re}\{J(f)\}\) in this class.

Theorem 6. In the subclass of all functions \(f(z)\) from \(\mathfrak{M}_{p,\alpha}(G)\) with any fixed value \(\operatorname{Re}\{J(f)\}\), the greatest exterior area in the domain \(G\) is attained by the function \(N_p(z,\zeta)+\lambda M_p(z,\zeta)\) of this subclass and, up to an additive constant, only by it.

For the case where the domain \(G\) is a circular annulus, the results listed above can be made explicit. Namely, one can find that for the circular annulus \(D: r<|z|<1,\ r>0\), we have:

\[ \varphi_{\theta,p}(z,\zeta)=Q_p\!\left(\frac{1}{z-\zeta}\right) +e^{2i\theta}\sum_{n=-\infty}^{\infty}{}' \frac{(\overline{\zeta}z)^n}{1-r^{2n}} \sum_{k=1}^{p}\binom{n-1}{k-1}\left(\frac{\alpha_k}{\zeta^k}\right) + \]

\[ +\sum_{n=1}^{\infty}\frac{r^{2n}}{1-r^{2n}} \sum_{k=1}^{p}\frac{\alpha_k}{\zeta^k} \left[ \binom{n-1}{k-1}\left(\frac{z}{\zeta}\right)^{-n} - \binom{-n-1}{k-1}\left(\frac{z}{\zeta}\right)^n \right]+c_0, \]

where \(c_0\) is a suitable constant.

From this we find explicit expressions for the functions \(M_p(z,\zeta)\) and \(N_p(z,\zeta)\) in the case of the annulus. For \(p=1\), the expressions obtained give the known functions \((^1)\). From Theorem 3, for the circular annulus there follows the theorem:

Theorem 7. Let \(f(z)\in\mathfrak{M}_{p,\alpha}(D)\) and
\[ f(z)-N_p(z,\zeta)=\sum_{n=-\infty}^{\infty} a_n z^n,\quad z\in D. \]

Then we have the sharp estimate:

\[ \sum_{n=-\infty}^{\infty} n(1-r^{2n})|a_n|^2 \leq \sum_{n=-\infty}^{\infty} \frac{n|\zeta|^{2n}}{1-r^{2n}} \left| \sum_{k=1}^{p}\binom{n-1}{k-1}\frac{\alpha_k}{\zeta^k} \right|^2 . \]

This theorem generalizes the corresponding theorem of Abe \((^6)\) for univalent functions in a circular annulus.

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

Received
4 IV 1962

References

\(^1\) Z. Nehari, Conformal Mapping, 1952.
\(^2\) H. Meschkowski, Ann. Acad. Sci. Fenn., Ser. A 1, 117 (1952).
\(^3\) Yu. E. Alenitsyn, Mat. sborn., 27, No. 2 (1950).
\(^4\) H. Grunsky, Math. Zs., 45, No. 1 (1939).
\(^5\) H. Grötzsch, Ber. d. Sächs. Akad. d. Wiss., Leipzig, 83, 283 (1931).
\(^6\) H. Abe, Math. Japon., 5, No. 1 (1958).