MATHEMATICS

I. E. BAZILEVICH and G. V. KORHITSKII

ON SOME PROPERTIES OF LEVEL LINES UNDER UNIVALENT CONFORMAL MAPPINGS

(Presented by Academician M. V. Keldysh on 22 IV 1961)

In a preceding joint paper \((^1)\) the authors proved that, under a univalent conformal mapping of the disk \(|z|<1\), realized by a function \(w=f(z)\) regular in this disk, the number of inflection points of the level lines \(L_r\) (the image of the circle \(|z|=r\)) and the number of points at which its starlikeness is violated (points of the level line at which the direction of rotation of the radius vector changes when \(z\) traverses the circle \(|z|=r\) in a fixed direction) may vary nonmonotonically as \(z\) approaches the boundary of the unit disk; i.e., if \(r_1<r_2\), it may turn out that the level line \(L_{r_1}\) has more inflection points or more points at which starlikeness is violated than the level line \(L_{r_2}\). The examples given in the cited paper consisted of functions mapping the disk \(|z|<1\) onto bounded domains.

The purpose of the present paper is to show that, when the modulus of a function univalent in the disk \(|z|<1\) grows sufficiently rapidly, a certain regularity must already be observed in the behavior of its level lines as \(r\to 1\). Namely, the following theorems hold.

Theorem 1. For the class \(S\) of functions
\[ f(z)=z+\sum_{n=2}^{\infty} c_n z^n, \]
regular and univalent in the disk \(|z|<1\), there exists an absolute constant \(\alpha_s\), \(0.1005<\alpha_s<0.134\), such that every arc of the level line \(L_r\) of any function \(f(z)\in S\), lying in the annulus
\[ \alpha_s\frac{r}{(1-r)^2}<|f(z)|<\frac{r}{(1-r)^2}, \qquad |z|=r<1, \]
is starlike; however, there exist functions \(f(z)\in S\) for which some arc of a level line lying in the wider annulus
\[ (\alpha_s-\varepsilon)\frac{r}{(1-r)^2}<|f(z)|<\frac{r}{(1-r)^2}, \qquad \varepsilon>0, \]
is no longer starlike, if \(r\) is sufficiently close to \(1\)*.

Theorem 2. For the class \(S\) of functions
\[ f(z)=z+\sum_{n=2}^{\infty} c_n z^n, \]
regular and univalent in the disk \(|z|<1\), there exists an absolute constant \(\alpha_k\), \(0.333\ldots<\alpha_k<0.511\), such that every arc of the level line \(L_r\) of any function \(f(z)\in S\), lying in the annulus
\[ \alpha_k\frac{r}{(1-r)^2}<|f(z)|<\frac{r}{(1-r)^2}, \qquad |z|=r<1, \]

* The quantity \(r/(1-r)^2\) is the exact upper bound for the modulus \(|f(z)|\), \(|z|\le r<1\), in the class \(S\).

is convex, but there exist functions \(f(z)\in S\) for which some arc of a level line lying in the wider annulus

\[ (a_k-\varepsilon)\frac{r}{(1-r)^2}<|f(z)|<\frac{r}{(1-r)^2}, \qquad \varepsilon>0, \]

will no longer be convex, if \(r\) is sufficiently close to \(1\).

Analogous theorems are proved for the class \(\Sigma\) of functions

\[ F(\zeta)=\frac{1}{f(1/\zeta)}=\zeta+\sum_{n=0}^{\infty}\frac{C_{-n}}{\zeta^n}, \qquad f(z)\in S,\qquad \zeta=\frac{1}{z}, \]

univalent and regular in the domain \(|\zeta|>1\), except for the simple pole \(\zeta=\infty\).

Theorem 3. For the class \(\Sigma\) of functions \(F(\zeta)=\dfrac{1}{f(1/\zeta)}\), \(f(z)\in S\), \(\zeta=1/z\), there exists an absolute constant \(A_s=1/a_s\), \(7.667<A_s<10\), such that every arc of the level line \(L_\rho\) \((\rho=|\zeta|)\) of any function \(F(\zeta)\in\Sigma\), lying in the annulus

\[ \frac{(\rho-1)^2}{\rho}<|F(\zeta)|<A_s\frac{(\rho-1)^2}{\rho}, \qquad |\zeta|=\rho>1, \]

is starlike; but there exist functions \(F(\zeta)\in\Sigma\) for which some arc of a level line lying in the wider annulus

\[ \frac{(\rho-1)^2}{\rho}<|F(\zeta)|<(A_s+\varepsilon)\frac{(\rho-1)^2}{\rho}, \qquad \varepsilon>0, \]

will no longer be starlike, if \(\rho\) is sufficiently close to \(1\)*.

Theorem 4. For the class \(\Sigma\) of functions \(F(\zeta)=\dfrac{1}{f(1/\zeta)}\), \(f(z)\in S\), \(\zeta=\dfrac{1}{z}\), there exists an absolute constant \(A_k\), \(1.75<A_k<10\), such that every arc of the level line \(L_\rho\) of any function \(F(\zeta)\in\Sigma\), lying in the annulus

\[ \frac{(\rho-1)^2}{\rho}<|F(\zeta)|<A_k\frac{(\rho-1)^2}{\rho}, \qquad |\zeta|=\rho>1, \]

is convex, but there exist functions \(F(\zeta)\in\Sigma\) for which some arc of a level line lying in the wider annulus

\[ \frac{(\rho-1)^2}{\rho}<|F(\zeta)|<(A_k+\varepsilon)\frac{(\rho-1)^2}{\rho}, \qquad \varepsilon>0, \]

will no longer be convex, if \(\rho\) is sufficiently close to \(1\).

Moscow Institute of Steel
named after I. V. Stalin

Received
22 IV 1961

CITED LITERATURE

1. I. E. Bazilevich, G. V. Koritskii, Mat. sborn., 32 (74), 1, 209 (1953).

* The quantity \((\rho-1)^2/\rho\) is the exact lower bound of the modulus \(|F(\zeta)|\), \(|\zeta|\ge \rho-1\), in the class \(\Sigma\).