UDC 517.54

MATHEMATICS

P. A. BILYUTA

EXTREMAL QUASICONFORMAL MAPPINGS

FOR ARBITRARY PLANE DOMAINS

(Presented by Academician M. A. Lavrent′ev on 21 I 1966)

In the present paper, by means of variational methods for conformal (1) and quasiconformal (2) mappings, the following extremal problem is solved.

We shall consider \(q\)-quasiconformal mappings \(w=f(z)\) of the disk \(|z|<1\) onto domains containing the points \(0,a,\ 0<a<1\), with normalization \(f(0)=0,\ f(a)=a\). Let
\(F(w_1,w_2,\ldots,w_n)\) be a real function of the variables
\(w_k=f(z_k)=u_k+iv_k\), continuously differentiable with respect to \(u_k,v_k\). It is required to find the mapping for which the function \(F\) assumes its maximum value for fixed values of \(z_k,\ k=1,2,\ldots,n\).

By virtue of the normality of the family of mappings under consideration, the extremal mapping exists.

In what follows we shall need the following

Theorem 1 (G. M. Goluzin (1)). If the function \(w=f(z),\ f(0)=0\), is regular and univalent in the disk \(|z|<1\), and the function \(w^*=\Phi(z,\lambda)\), as a function of \(z\) and \(\lambda\), is regular for \(|\lambda|<\lambda_0\) and \(r\le |z|<1\), and for every \(\lambda,\ 0<\lambda<\lambda_0\), it is univalent in \(r\le |z|<1\); if, moreover, for every \(z,\ r\le |z|<1\), and small \(\lambda\) we have

\[ \Phi(z,\lambda)=f(z)+\lambda q(z)+O(\lambda^2), \]

then, adjoining to the image of the annulus \(r<|z|<1\) under the mapping by the function \(w^*=\Phi(z,\lambda)\) the domain internal with respect to the image of the circle \(|z|=r\), for small \(\lambda\) we obtain a simply connected domain \(D^*\), containing the point \(w^*=0\), and for the function \(w^*=f^*(z),\ f^*(0)=0\), univalently mapping the disk \(|z|<1\) onto \(D^*\), we have, for \(|z|<1\),

\[ f^*(z)=f(z)+\lambda q(z)-\lambda z f'(z)S(z)+\lambda z f'(z)\overline{S(1/\bar z)}+O(\lambda^2), \tag{1} \]

where \(S(z)\) is the sum of the terms with negative powers of \(z\) in the expansion of \(q(z)/zf'(z)\) in the annulus \(r<|z|<1\).

The solution of the extremal problem stated above is given by

Theorem 2. The function \(w=f(z)\) for which \(F(w_1,w_2,\ldots,w_n)\) attains its maximum for fixed values \(z_k,\ k=1,2,\ldots,n\), has the following properties:

1) the mapping carried out by the function inverse to the extremal one has characteristics

\[ p(w)=q,\qquad \theta(w)=-\frac{1}{2}\arg A(w), \]

where

\[ A(w)=\sum_{k=1}^{n} \frac{F_{w_k} w_k(a-w_k)} {\bar w(a-w)(w_k-w)} \]

(we leave aside the case \(A\equiv0\), corresponding to the presence of a stationary value of \(F\): \(F_{w_k}=0,\ k=1,2,\ldots,n\));

2) it maps the disk \(|z|<1\) onto the whole \(w\)-plane with cuts along a finite number of analytic arcs satisfying the inequality

\[ A(w)\,dw^2>0. \tag{*} \]

\(1^\circ\). Let \(w=f(z)\) be an extremal mapping. Subjecting the \(w\)-plane to a variation with constant characteristic \(h\) in the disk \(K(\zeta,r): |w-\zeta|\le r\), according to the formula

\[ \omega=w+hr^2\frac{w(a-w)}{\zeta(a-\zeta)(w-\zeta)} \quad \text{for } |w-\zeta|>r, \tag{2} \]

\[ \omega=w+h(\overline{w}-\overline{\zeta}) \quad \text{for } |w-\zeta|\le r \]

(obviously, this variation does not take us out of the class of domains under consideration), we find that

\[ \delta p=-2p|h|\cos 2(\theta^*-\theta), \tag{3} \]

\[ dF=-2r^2|hA|\cos 2(\theta^*-\theta_A), \tag{4} \]

where \(\theta^*=\tfrac12\arg h+\pi/2\), \(\theta_A=-\tfrac12\arg A(\zeta)\); \(p,\theta\) are the characteristics of the mapping inverse to \(w=f(z)\), at the point \(w=\zeta\).

From (3) and (4) the first assertion of Theorem 2 follows immediately.

\(2^\circ\). From (4) we also conclude that the extremal domain \(D\) cannot have exterior points, since otherwise any variation (2) with constant characteristic \(h\) in a disk lying outside the domain would be admissible, and \(dF\) could have any sign.

Represent the extremal mapping \(w=f(z)\), \(f(0)=0\), \(f(a)=a\), in the form of a superposition of two mappings: 1) a quasiconformal mapping \(\zeta=g(z)\) of the disk \(|z|<1\) onto the disk \(|\zeta|<1\), \(g(0)=0\), \(g(a)=b\), where \(b\), \(0<b<1\), is a certain number, with characteristics \(p(z)=q\), \(\theta(z)\) of the mapping \(w=f(z)\); and 2) a conformal mapping \(w=\varphi(\zeta)\) of the disk \(|\zeta|<1\) onto the domain \(D\), with \(\varphi(0)=0\), \(\varphi(b)=a\).

First consider the function

\[ w^*=w+hw/w_0(w-w_0), \tag{5} \]

where \(w_0\) is an arbitrary finite point of the \(w\)-plane. For any prescribed \(\rho>0\) and sufficiently small complex \(h\), the function (5) is univalent in the infinite domain \(|w-w_0|>\rho\).

Now let the function \(w=\varphi(\zeta)\), \(\varphi(0)=0\), \(\varphi(b)=a\), be regular in the disk \(|\zeta|<1\) and map \(|\zeta|<1\) univalently onto the domain \(D\). If \(w_0\in D\) and \(\zeta_0\) is from \(|\zeta|<1\) such that \(w_0=\varphi(\zeta_0)\), then for sufficiently small \(h\) the function

\[ w^*=\varphi(\zeta)+h\varphi(\zeta)/\varphi(\zeta_0)\bigl(\varphi(\zeta)-\varphi(\zeta_0)\bigr) \]

will be regular and univalent in some annulus \(r\le |\zeta|<1\). For \(\lambda=|h|\), Theorem 1 is applicable to this function. Since the function

\[ \frac{q(\zeta)}{\zeta\varphi'(\zeta)} = e^{i\alpha}\frac{\varphi(\zeta)} {\zeta\varphi'(\zeta)\varphi(\zeta_0)\bigl(\varphi(\zeta)-\varphi(\zeta_0)\bigr)}, \quad \alpha=\arg h, \]

in the disk \(|\zeta|<1\) has only a simple pole at the point \(\zeta_0\) with residue \(e^{i\alpha}/\zeta_0\varphi'(\zeta_0)^2\), it is clear that

\[ S(\zeta)=e^{i\alpha}/\zeta_0\varphi'(\zeta_0)^2(\zeta-\zeta_0), \]

and, consequently, formula (1) gives

\[ \varphi^*(\zeta)=\varphi(\zeta) +h\frac{\varphi(\zeta)}{\varphi(\zeta_0)\bigl(\varphi(\zeta)-\varphi(\zeta_0)\bigr)} -h\frac{\zeta\varphi'(\zeta)}{\zeta_0\varphi'(\zeta_0)^2(\zeta-\zeta_0)} + \]

\[ +\overline{h}\frac{\zeta^2\varphi'(\zeta)} {\overline{\zeta_0\varphi'(\zeta_0)^2}(1-\overline{\zeta_0}\zeta)} +O(|h|^2). \tag{6} \]

Let us further normalize the varied function (6), putting \(\omega=a\varphi^*(\zeta)/\varphi^*(b)\), after which we obtain

\[ \begin{aligned} \omega=\varphi(\zeta) &+h\,\frac{\varphi(\zeta)(a-\varphi(\zeta))} {\varphi(\zeta_0)(a-\varphi(\zeta_0))(\varphi(\zeta)-\varphi(\zeta_0))} -h\,\frac{\zeta\varphi'(\zeta)} {\zeta_0\varphi'(\zeta_0)^2(\zeta-\zeta_0)} \\ &+h\,\frac{b\varphi'(b)\varphi(\zeta)} {a\zeta_0\varphi'(\zeta_0)^2(b-\zeta_0)} +\bar h\,\frac{\zeta^2\varphi'(\zeta)} {\zeta_0\varphi'(\zeta_0)^2(1-\bar\zeta_0\zeta)} -\bar h\,\frac{b^2\varphi'(b)\varphi(\zeta)} {a\zeta_0\varphi'(\zeta_0)^2(1-b\bar\zeta_0)} +O(|h|^2). \end{aligned} \tag{7} \]

This is the variational formula for functions \(w=\varphi(\zeta)\) mapping conformally the disk \(|\zeta|<1\) onto the domain \(D\) with normalization \(\varphi(0)=0,\ \varphi(b)=a\).

If now \(w=\varphi\circ g\) is an extremal mapping, then, varying the function \(\varphi(\zeta)\) according to formula (7), we obtain

\[ \begin{aligned} dF=2\operatorname{Re}h\sum_{k=1}^{n}\Bigg[ &\frac{F_{w_k}w_k(a-w_k)} {\varphi(\zeta_0)(a-\varphi(\zeta_0))(w_k-\varphi(\zeta_0))} -\frac{F_{w_k}\zeta_k w'_k} {\zeta_0\varphi'(\zeta_0)^2(\zeta_k-\zeta_0)} \\ &+\frac{F_{w_k}w_k b\varphi'(b)} {a\zeta_0\varphi'(\zeta_0)^2(b-\zeta_0)} +\frac{\overline{F}_{w_k}\bar\zeta_k^{\,2}\bar w'_k} {\zeta_0\varphi'(\zeta_0)^2(1-\zeta_0\bar\zeta_k)} +\frac{\overline{F}_{w_k}\bar w_k b^2\varphi'(b)} {a\zeta_0\varphi'(\zeta_0)^2(1-b\bar\zeta_0)} \Bigg]\leq 0, \end{aligned} \]

where

\[ \zeta_k=g(z_k),\qquad w_k=\varphi(\zeta_k),\qquad w'_k=\varphi'(\zeta_k). \]

By virtue of the arbitrariness of \(\arg h\) and \(\zeta_0\), we conclude that in the case of an extremal mapping the function \(\varphi(\zeta)\) satisfies the differential equation

\[ \frac{\zeta\varphi'(\zeta)^2}{\varphi(\zeta)(a-\varphi(\zeta))} \sum_{k=1}^{n}\frac{F_{w_k}w_k(a-w_k)}{1-\bar\zeta\zeta_k} = \]

\[ = \sum_{k=1}^{n} \left[ \frac{F_{w_k}\zeta_k w'_k}{\zeta_k-\zeta} -\frac{F_{w_k}w_k b\varphi'(b)}{a(b-\zeta)} -\frac{\overline{F}_{w_k}\bar\zeta_k^{\,2}\bar w'_k}{1-\zeta\bar\zeta_k} +\frac{\overline{F}_{w_k}\bar w_k b^2\varphi'(b)}{a(1-b\zeta)} \right]. \]

From the analytic theory of differential equations it follows that \(\varphi(\zeta)\) is regular not only in the disk \(|\zeta|<1\), but also on the circumference \(|\zeta|=1\), except for a finite number of points. Consequently, the boundary of the domain \(D\) consists of a finite number of analytic arcs.

\(3^\circ\). Let us prove that the slits forming the boundary of the extremal domain \(D\) satisfy inequality \((*)\).

Suppose, to the contrary, that at some point of a slit \(w=w_0\), distinct from its end, the angle between the direction \(dw_0=-\frac12\arg A(w_0)\) and the slit is equal to \(\theta,\ 0<\theta<\pi\). In view of the analyticity of the boundary and of the fact that we shall vary the \(w\)-plane in a sufficiently small neighborhood of the point \(w_0\), one may regard the arc of the slit lying in this neighborhood as a rectilinear segment; one may also take \(w_0=0\).

We now vary the \(w\)-plane as follows. Outside the segment

\[ u^2+(v+c\operatorname{ctg}\alpha)^2<c^2/\sin^2\alpha,\qquad v>0 \]

the mapping is identical, and we contract the segment \(1/k\) times in the direction \(\theta\); here \(c>0\) is sufficiently small, while \(\alpha,\ 0<\alpha<\pi/2\), and \(k,\ 0<k<1\), will be chosen later. This mapping \(\omega=\omega(w)\), \(\omega=\xi+i\eta\), has the form

\[ \eta=(k\sin^2\theta+\cos^2\theta)v -(1-k)\sin\theta\,[u\cos\theta+c\operatorname{ctg}\alpha\sin\theta- \]

\[ -\sqrt{c^2(1+\operatorname{ctg}^2\alpha) -(v\cos\theta-u\sin\theta+c\operatorname{ctg}\alpha\cos\theta)^2}], \]

\[ \xi=(\eta-v)\operatorname{ctg}\theta+u. \]

The condition that, under the additional mapping \(\omega=\omega(w)\), the characteristic \(p\) decreases is as follows:

\[ (\xi_u^2+\eta_u^2-\xi_v^2-\eta_v^2)\cos 2\theta +2(\xi_u\xi_v+\eta_u\eta_v)\sin 2\theta<0. \tag{8} \]

For small \(\alpha\) we have

\[ \xi_u=1+O(\alpha),\qquad \xi_v=-(1-k)\operatorname{ctg}\theta+O(\alpha), \]

\[ \eta_u=O(\alpha),\qquad \eta_v=k+O(\alpha), \]

and condition (8) is written as

\[ -(1-k)\,[2+(1-k)\cos 2\theta/\sin^2\theta]+O(\alpha)<0. \]

For any \(\theta,\ 0<\theta<\pi\), this inequality can be satisfied if \(k\) is taken so close to 1 that the expression in square brackets is positive, and \(\alpha\) is sufficiently small.

Thus, we have constructed a mapping that has not changed the value of \(F\), but maps onto a domain with exterior points. This contradiction proves our assertion, which, together with the results of item \(2^\circ\), completes the proof of Theorem 2.

Remark. Obviously, the same extremal problem can be considered for \(q\)-quasiconformal mappings \(w=f(z)\) of an arbitrary simply connected domain \(D\) of the \(z\)-plane with the normalization \(f(z')=0,\ f(z'')=a\), where \(z',z''\in D\), and in this case Theorem 2 holds for the extremal function.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
12 I 1966

CITED LITERATURE

1. G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Moscow, 1952.
2. P. P. Belinskii, DAN, 121, No. 2 (1958).