MATHEMATICS

Yu. D. MAKSIMOV

A GENERALIZATION OF THE STRUCTURAL FORMULA FOR CONVEX UNIVALENT FUNCTIONS TO THE CASE OF A MULTIPLY CONNECTED CIRCULAR DOMAIN

(Presented by Academician V. I. Smirnov, 1 VIII 1960)

The well-known structural formula for the class of convex univalent functions regular in the disk \(|z| < 1\) was generalized by V. A. Zmorovich to the case of the annulus \(q < |z| < 1\) (\(^1\)). In the present note a further generalization of this formula is given for the case of a multiply connected circular domain—the structural formula for the class \(B(D_{n+1})\) of functions \(f(z)\), regular and univalent in the domain \(D_{n+1}\), bounded from the outside by the circle \(|z-a_0|=r_0\) and from the inside by the circles \(|z-a_k|=r_k\); \((k=1,2,\ldots,n)\); \(|a_k-a_l|>r_k+r_l;\ k,l=1,2,\ldots,n;\ k\ne l\), mapping \(D_{n+1}\) onto domains bounded by \(n+1\) convex lines.

The solution is given in constructive form, but not for all domains \(D_{n+1}\), only for those satisfying certain conditions characterizing the convergence of the series involved in the formation of the structural formula. In its construction a method was used that had been employed by G. M. Goluzin in solving Dirichlet and Neumann type problems (\(^2\)). We note that in the work of L. O. Dunduchenko and S. A. Kasyanyuk (\(^3\)) a construction is carried out of a structural formula for the class of functions mapping the \(z\)-plane with \(n\) disks \(|c_k-z|\le R_k\ (k=1,2,\ldots,n)\) removed onto domains of bounded boundary rotation in the sense of Paatero (\(^4\)), in particular onto domains bounded by convex lines and polygons. However, that work contains an error, and as a result the structural formula turns out to be incorrect. It appears impossible to solve this problem by the method used by the authors mentioned.

Denote by \(C_m\) the circle \(|z-a_m|=r_m\ (m=0,1,\ldots,n)\). Let \(C_m^\rho\) be a circle concentric with \(C_m\), lying wholly in \(D_{n+1}\) and at distance \(\rho\) from \(C_m\).

Consider the auxiliary class \(K(D_{n+1})\) of functions \(\varphi(z)\) satisfying the following conditions:

1) \(\varphi(z)\) is regular in \(D_{n+1}\) and can be represented in the form

\[ \varphi(z)=b_0^{(0)}+b_1^{(0)}(z-a_0)+b_2^{(0)}(z-a_0)^2+\cdots \]

\[ \cdots+\sum_{m=1}^{n}\left[\frac{b_2^{(m)}}{(z-a_m)^2}+\frac{b_3^{(m)}}{(z-a_m)^3}+\cdots\right]. \tag{1} \]

2) On the circles \(C_m^\rho\ (m=0,1,\ldots,n)\), for sufficiently small \(\rho\),

\[ \int_{-\pi}^{\pi}\alpha_{m,\rho}^{-}(\theta)\,d\theta<\varepsilon \tag{2} \]

whatever \(\varepsilon>0\) may be. Here \(\theta=\arg(z-a_m)\); \(z\in C_m^\rho\);

\[ \alpha_{m,\rho}^{-}(\theta)= \begin{cases} -\operatorname{Re}[1+(z-a_m)\varphi(z)], & \text{if } \operatorname{Re}[1+(z-a_m)\varphi(z)]\le 0,\\ 0, & \text{if } \operatorname{Re}[1+(z-a_m)\varphi(z)]>0. \end{cases} \]

Theorem 1. The structural formula

\[ \varphi(z)=\frac{1}{2\pi}\sum_{k=0}^{n}\int_{-\pi}^{\pi}\Phi_k(z,\theta)\,d\mu_k(\theta), \tag{3} \]

where \(\mu_k(\theta)\) are nondecreasing on \([-\pi,\pi)\) and

\[ \int_{-\pi}^{\pi}d\mu_k(\theta)=2\pi \quad (k=0,1,2,\ldots,n), \]

is a necessary and sufficient condition for the function \(\varphi(z)\) to belong to the class \(k(D_{n+1})\). Here

\[ \Phi_k(z,\theta)= \sum_{\nu=1}^{\infty} \left[ h_{\nu 0 k}(z,\theta)+ \sum_{m=1}^{n}\frac{h_{\nu m k}(z,\theta)}{(z-a_m)^2} \right] + \frac{F_k(z,\theta)}{(z-a_k)^2} \quad (k=1,2,\ldots,n); \]

\[ \Phi_0(z,\theta)= \sum_{\nu=1}^{\infty} \left[ h_{\nu 0 0}(z,\theta)+ \sum_{m=1}^{n}\frac{h_{\nu m 0}(z,\theta)}{(z-a_m)^2} \right] + F_0(z,\theta); \tag{4} \]

\[ F_0(z,\theta)= \frac{2e^{-i\theta}}{r_0-(z-a_0)e^{-i\theta}}; \qquad F_k(z,\theta)= \frac{-2r_k(z-a_k)}{r_k-(z-a_k)e^{-i\theta}}; \tag{5} \]

for \(\nu\) odd,

\[ h_{\nu m k}(z,\theta)= \sum_{k_{\nu-1},\ldots,k_1} B_{m k_{\nu-1}}(z)\, \overline{B_{k_{\nu-1}k_{\nu-2}}([m,z])}\cdots \]

\[ \cdots B_{k_2 k}([k_2,\ldots,m,z])\, \overline{F_k([k_1,k_2,\ldots,k_{\nu-1},m,z],\theta)}; \tag{6} \]

for \(\nu\) even,

\[ h_{\nu m k}(z,\theta)= \sum_{k_{\nu-1},\ldots,k_1} B_{m k_{\nu-1}}(z)\, \overline{B_{k_{\nu-1}k_{\nu-2}}([m,z])}\cdots \]

\[ \cdots \overline{B_{k_1 k}([k_2,\ldots,m,z])}\, F_k([k_1,k_2,\ldots,k_{\nu-1}m,z],\theta). \]

In formulas (6), \(k,m=0,1,\ldots,n\); the summation over \(k_1,\ldots,k_{\nu-1}\) is carried out from \(0\) to \(n\);

\[ B_{mk}(z)= \frac{-r_m^2}{([m,z]-\bar a_k)^2}, \qquad m,k=1,2,\ldots,n;\quad m\ne k; \]

\[ B_{0k}(z)= \frac{-r_0^2}{[(\bar a_0-\bar a_k)(z-a_0)+r_0^2]^2}, \qquad k=1,2,\ldots,n; \tag{7} \]

\[ B_{m0}(z)=-r_m^2, \qquad m=1,2,\ldots,n; \]

\[ B_{kk}(z)=0, \qquad k=0,1,\ldots,n; \]

for \(\nu\) odd,

\[ [m,z]=a_m+\frac{r_m^2}{z-\bar a_m}; \]

\[ [k_1,k_2,\ldots,k_{\nu-1},k_\nu,z]= \]

\[ = a_{k_1} +\frac{r_{k_1}^2}{ -a_{k_1}+\bar a_{k_2} +\dfrac{r_{k_2}^2}{ -a_{k_2}+\bar a_{k_3} +\cdots +\dfrac{r_{k_{\nu-1}}^2}{ -a_{k_{\nu-1}}+\bar a_\nu +\dfrac{r_\nu^2}{-\bar a_\nu+z} } } }; \]

for even \(\nu\)

\[ [k_1,k_2,\ldots,k_{\nu-1},k_\nu,z]= \]

\[ = a_{k_1}+\frac{r_{k_1}^2}{-a_{k_1}+a_{k_2}+\dfrac{r_{k_2}^2}{-a_{k_2}+a_{k_3}+ \cdots +\dfrac{r_{k_{\nu-1}}^2}{-a_{k_{\nu-1}}+a_\nu+\dfrac{r_\nu^2}{-a_\nu+z}}}} . \]

Let

\[ \delta_m=\left[1+\sum_{\substack{s=1\\ s\ne m}}^n \frac{1}{|a_m-a_s|-r_m}\right]r_m^2,\qquad m=1,2,\ldots,n; \]

\[ \delta_0=\sum_{s=1}^n \frac{1}{\left[r_0-|a_0-a_s|\right]^2}. \]

Theorem 2. If the conditions

\[ \delta_m<1,\qquad m=0,1,2,\ldots,n, \tag{8} \]

are satisfied, then the series (4) converge absolutely and uniformly in the domain \(D_{n+1}\).

These conditions, roughly speaking, will be satisfied when \(r_m\) \((m=1,2,\ldots,n)\) are sufficiently small, and \(r_0\) is sufficiently large.

Main theorem. The structural formula

\[ f(z)=\int_{z_0}^{z}\exp\left[\int_{z_1}^{z}\varphi(z)\,dz\right]dz, \tag{9} \]

where \(z_0,z_1\) are fixed points of \(D_{n+1}\), \(\varphi(z)\in K(D_{n+1})\), is a necessary and sufficient condition for the function \(f(z)\) to belong to the class \(B(D_{n+1})\) under the normalization \(f(z_0)=0,\ f'(z_1)=1\). The function \(\varphi(z)\) is chosen so that all residues of the integrand
\[ \exp\left[\int_{z_1}^{z}\varphi(z)\,dz\right] \]
at the points \(a_k\) \((k=1,2,\ldots,n)\) are equal to zero.

Corollary 1. If it is assumed that the functions \(\mu_k(\theta)\) entering formula (9) are step functions with points of discontinuity \(\theta_{k j_k}\) \((k=0,1,\ldots,n;\ j_k=1,2,\ldots,\nu_k;\ -\pi<\theta_{k1}<\theta_{k2}<\cdots<\theta_{k\nu_k}<\pi)\) and jumps at these points \(2\pi\sigma_{k j_k}\) \(\left(\sigma_{k j_k}>0;\ \sum_{j_k=1}^{\nu_k}\sigma_{k j_k}=1\right)\), then formula (9) gives regular, univalent in \(D_{n+1}\) functions \(f(z)\), mapping \(D_{n+1}\) onto domains bounded by convex polygons with vertices \(f(r_k e^{i\theta_{k j_k}})\) and with interior angles at them equal to \(\pi(1-2\sigma_{k j_k})\).

The mapping function in this case has the form (without normalization)

\[ f(z)=c\int_{z_0}^{z}\prod_{k=0}^{n}\prod_{j_k=1}^{\nu_k}\Psi_{k j_k}(z)\,dz+c_1, \tag{10} \]

where

\[ \Psi_{k j_k}(z)=\exp\left[\sigma_{k j_k}\int_{z_1}^{z}\Phi_k(z,\theta_{k j_k})\,dz\right]. \tag{11} \]

Corollary 2. If in formula (11) the \(\sigma_{k j_k}\) are of arbitrary sign, but remain subject to the conditions

\[ \sum_{j_k=1}^{\nu_k}\sigma_{k j_k}=1\qquad (k=0,1,\ldots,n), \]

then formula (10) will give functions mapping \(D_{n+1}\) onto domains bounded by polygons (not necessarily convex or even univalent), and formula (10) ensures that \(f'(z)\ne 0\), i.e., the absence of branch points in the image. However, in this case the mapping will, generally speaking, be non-univalent. Formula (10) in this case represents a generalization of the well-known Christoffel–Schwarz formula to the case of a multiply connected circular domain. In particular, for \(n=2\), in the case of a concentric circular annulus we obtain the Tiri–Akhiezer–Goluzin formula \((^1)\). For \(n=1\) we have the Christoffel–Schwarz formula for the disk.

Petropavlovsk-Kamchatsky
Pedagogical Institute

Received
27 VI 1960

REFERENCES

\(^1\) V. A. Zmorovich, Matem. sborn., 32, no. 3, 633 (1953).
\(^2\) G. M. Goluzin, Matem. sborn., 41, 246 (1934).
\(^3\) L. O. Dunduchenko, S. A. Kasyanyuk, Dopovid. AN URSR, 3, 227 (1959).
\(^4\) V. Paatero, Ann. Acad. Sci. Fennicae, ser. A, 147 (1953).