MATHEMATICS

G. K. ANTONYUK

COVERING OF AREAS FOR FUNCTIONS REGULAR IN AN ANNULUS

(Presented by Academician V. I. Smirnov, 22 XI 1956)

A function \(w=f(z)\in\mathfrak M\) if in the annulus \(1<|z|<R\) \(w=f(z)\) is regular and

\[ \frac{1}{2\pi i}\int_L \frac{f'(z)}{f(z)}\,dz \geqslant 1, \]

where \(L\) is a contour not homologous to zero.

The question of covering areas for functions of the class \(\mathfrak M\) was posed by G. Ya. Khazaliya in the papers \((^{1,2})\). In paper \((^1)\) the problem of minimizing the area of the image obtained under the mapping of the annulus \(1<|z|<R\) by a function \(w=f(z)\in\mathfrak M\) is solved. In paper \((^2)\) the notion of a star of the Riemann surface onto which the annulus \(1<|z|<R\) is mapped by a function \(w=f(z)\in\mathfrak M\) is introduced, and its area \(S^*\) is estimated: \(S^*\geqslant \pi(R^2-1)\). Unfortunately, the definition of a star has shortcomings, which, in particular, were pointed out in \((^3)\); moreover, the existence of a star was not proved. We give a somewhat different definition of a star (cf. \((^{2,4})\)).

1. Let \(\mathfrak R\) and \(\mathfrak R_{\tau,\sigma}\) be Riemann surfaces onto which, respectively, the annuli \(1<|z|<R\) and \(1+\tau<|z|<R-\sigma\), \(\tau,\sigma>0\), are mapped by the function \(w=f(z)<\mathfrak M\); let \(\Gamma_\tau,\Gamma_\sigma\) be the images of the circles \(|z|=1+\tau\) and \(|z|=R-\sigma\), obtained under this mapping. Let two arbitrary sequences of numbers \(\tau_n\to0\) and \(\sigma_n\to0\), \(n\to\infty\), be given, satisfying the conditions: \(\tau_k>\tau_{k+1}\), \(\sigma_k>\sigma_{k+1}\), \(k=1,2,\ldots\). Let an arbitrary system of rays \(L_k\), \(k=1,2,\ldots,m\), drawn from the point \(w=0\), be given.

2. We cut the Riemann surface \(\mathfrak R_{\tau_1,\sigma_1}\) over all its sheets:

a) by the system of rays \(L_k\), \(k=1,2,\ldots,m\);

b) by the system of rays drawn from the point \(w=0\) through all branch points of \(\overline{\mathfrak R_{\tau_1,\sigma_1}}\);

c) by the system of all rays issuing from the point \(w=0\) and touching the boundary of the Riemann surface \(\mathfrak R_{\tau_1,\sigma_1}\).

The same rays cut each of the Riemann surfaces \(\mathfrak R_{\tau_2,\sigma_2}\), \(\mathfrak R_{\tau_3,\sigma_3},\ldots\) into a certain totality of parts. Let \(\omega_1,\omega_2,\ldots,\omega_s\) be the sectors formed by neighboring rays of the indicated subdivision, and let their apertures be equal, respectively, to \(\alpha_1,\alpha_2,\ldots,\alpha_s\). Among the parts of the Riemann surfaces obtained under the indicated subdivision and contained in the sector \(\omega_q\), \(q=1,2,\ldots,s\), we single out all possible sequences of them (let their number be \(k_q\)*) of the form

\[ \mathfrak R_{\tau_1,\sigma_1}^{q,j}\subset \mathfrak R_{\tau_2,\sigma_2}^{q,j}\subset \mathfrak R_{\tau_3,\sigma_3}^{q,j}\subset \ldots,\quad j=1,2,\ldots,k_q\ (k_q\geqslant 1),\quad 1\leqslant q\leqslant s, \]

where \(\mathfrak R_{\tau_k,\sigma_k}^{q,j}\) is such a part of the Riemann surface \(\mathfrak R_{\tau_k,\sigma_k}\), \(k=1,2,\ldots\), lying in the sector \(\omega_q\), that the increment of \(\arg w\) along all pieces \(\Gamma_{\tau_k},\Gamma_{\sigma_k}\) entering the boundary of \(\mathfrak R_{\tau_k,\sigma_k}^{q,j}\), when they are traversed in the positive direction, is respectively equal to \(-\alpha_q\) and \(+\alpha_q\).

1. On the Riemann surface \(\mathfrak R_{\tau_2,\sigma_2}\), \(1\leqslant q\leqslant s\), \(1\leqslant j\leqslant k_q\), mark:

a) all branch points of \(\mathfrak R_{\tau_2,\sigma_2}^{q,j}\), whose projection onto the \(w\)-plane lies outside the projection of \(\mathfrak R_{\tau_1,\sigma_1}^{q,j}\);

* It is clear that the number \(k_q\) of sequences depends on the structure of the Riemann surface.

b) all those points of the boundary of the Riemann surface \(\mathfrak R_{\tau_2,\sigma_2}^{q,j}\) which have projection onto the \(w\)-plane outside the projection of \(\mathfrak R_{\tau_1,\sigma_1}^{q,j}\), at which the rays drawn from the point \(w=0\) touch the boundary of \(\mathfrak R_{\tau_2,\sigma_2}^{q,j}\).

From the point \(w=0\), through the marked points, draw a system of rays. Cut the Riemann surface \(\mathfrak R_{\tau_2,\sigma_2}^{q,j}\) along all its sheets in the following way: draw a cut, not passing along \(\mathfrak R_{\tau_1,\sigma_1}^{q,j}\), from each of the marked points along the corresponding ray to \(w=\infty\) or \(w=0\). Among the parts of the Riemann surface obtained under such a partition, choose the one which contains \(\mathfrak R_{\tau_1,\sigma_1}^{q,j}\); denote it by \(V_{\tau_2,\sigma_2}^{q,j}\). On the Riemann surface \(\mathfrak R_{\tau_3,\sigma_3}^{q,j}\), by means of \(V_{\tau_2,\sigma_2}^{q,j}\), mark all points of the same type as the points marked by us on \(\mathfrak R_{\tau_2,\sigma_2}^{q,j}\) by means of \(\mathfrak R_{\tau_1,\sigma_1}^{q,j}\). Add to them the points marked by us earlier on \(\mathfrak R_{\tau_2,\sigma_2}^{q,j}\). Separate from the surface \(\mathfrak R_{\tau_3,\sigma_3}^{q,j}\), by means of \(V_{\tau_2,\sigma_2}^{q,j}\), the part \(V_{\tau_3,\sigma_3}^{q,j}\) by partitioning it along the rays drawn from the point \(w=0\) through the resulting set of points, just as we separated \(V_{\tau_2,\sigma_2}^{q,j}\) from \(\mathfrak R_{\tau_2,\sigma_2}^{q,j}\) by means of \(\mathfrak R_{\tau_1,\sigma_1}^{q,j}\).

Continuing in the same way the partition of the Riemann surfaces
\(\mathfrak R_{\tau_4,\sigma_4}^{q,j}\), \(\mathfrak R_{\tau_5,\sigma_5}^{q,j}\), \(\ldots\), we obtain a sequence of their parts:

\[ V_{\tau_1,\sigma_1}^{q,j}=\mathfrak R_{\tau_1,\sigma_1}^{q,j}\subset V_{\tau_2,\sigma_2}^{q,j}\subset V_{\tau_3,\sigma_3}^{q,j}\subset\ldots . \]

In this case the parts \(V_{\tau_k,\sigma_k}^{q,j}\), \(k=1,2,\ldots\), are simply connected; every ray drawn from the point \(w=0\) through a point of \(V_{\tau_k,\sigma_k}^{q,j}\) has only one segment which is a cross-section of \(V_{\tau_k,\sigma_k}^{q,j}\); the increment of \(\arg w\) along all arcs \(\Gamma_{\tau_k}\), \(\Gamma_{\sigma_k}\) entering into the boundary of \(V_{\tau_k,\sigma_k}^{q,j}\), when they are traversed in the positive direction, is respectively equal to \(-\alpha_q\) and \(+\alpha_q\).

1. Let \(j_q\) \((q=1,2,\ldots,s)\) be some arbitrary number among the numbers \(1,2,\ldots,k_q\). Denote by \(V_n(j_1,\ldots,j_s)\) the totality of the domains

\[ \sum_{q=1}^{s} V_{\tau_n,\sigma_n}^{q,j_q},\qquad n=1,2,\ldots . \]

The star \(\mathfrak R^*\) of the Riemann surface \(\mathfrak R\) with respect to the system \(L_k\), \(k=1,2,\ldots,m\), of rays drawn from the point \(w=0\), is the open set situated on the Riemann surface \(\mathfrak R\) and which is the limiting set for some sequence \(V_n(j_1,j_2,\ldots,j_s)\) as \(n\to\infty\) (it is clear that if at least one of the numbers \(k_1,k_2,\ldots,k_s\) is greater than one, then the star is not unique).

It is evident that the star \(\mathfrak R^*\) is simply connected and that a ray drawn from the point \(w=0\) through a point of \(\mathfrak R^*\) has only one segment, lying in \(\mathfrak R^*\), with endpoints on the boundary of \(\mathfrak R^*\).

The star defined in the indicated manner exists. Using the method of A. F. Bermant \(({}^5)\), p. 192, one can obtain a result more general than the result of G. Ya. Khazalia:

\[ \left(1+\frac{S}{\pi}\right)\left(1+\frac{s}{\pi}\right)\geq R^4, \]

where \(S\) is the area of the star, \(s\) is the area of its transform; equality is possible only for the function \(f(z)=\varepsilon z\), \(|\varepsilon|=1\).

Received
20 XI 1956

CITED LITERATURE

\({}^{1}\) G. Ya. Khazalia, DAN, 20, No. 2–3 (1938).
\({}^{2}\) G. Ya. Khazalia, Tr. Matem. inst. im. Razmadze, AN GruzSSR, 18, 245 (1951).
\({}^{3}\) Math. Rev., 14, No. 6, 549 (1953).
\({}^{4}\) A. F. Bermant, Matem. sborn., 20, 55 (1947).
\({}^{5}\) G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, 1952.