Abstract
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Reports of the Academy of Sciences of the USSR
- Volume 163, No. 6
MATHEMATICS
O. V. STEPANOVA
ON ONE PROPERTY OF LEVEL LINES UNDER UNIVALENT CONFORMAL MAPPINGS
(Presented by Academician M. A. Lavrent'ev, February 27, 1965)
Let \(S\) be the class of regular and univalent functions
\[ f(x)=z+a_2z^2+a_3z^3+\cdots \]
and let \(L(f,r)\) be the image of the circle \(|z|=r\) under its mapping in the disk \(|z|<1\) by the function \(f(z)\) (a level line). In papers \((^{1-3})\) the connection was investigated between the property of the level line \(L(f,r)\) of being star-shaped or convex and the value of the modulus of the function \(f(z)\). In the present work a connection is established between the star-shapedness of an arc of a level line and the value of the modulus of the derivative \(f'(z)\). For this purpose, on the basis of the method set forth in paper \((^2)\), the domain \(D_r\) of values of the functional
\[ I(f)=\ln |f'(z)|-i\arg\bigl(zf''(z)/f'(z)\bigr),\qquad |z|=r, \]
in the class \(S\), is found.
The domain \(D_r\) is closed, bounded, convex, and symmetric with respect to the real axis. In the upper half-plane it is bounded by a continuous curve, whose endpoints lie on the real axis and which consists of arcs of four analytic curves. For \(r>(\sqrt3-1)/\sqrt2\), among these arcs there is a line segment; for \(r\le(\sqrt3-1)/\sqrt2\) the segment contracts to a point. The equations of all these arcs have been found in explicit analytic form.
The investigation carried out of the domain of values of the functional \(I(f)\) makes it possible to formulate the following theorem:
Theorem. For each \(r\) there exist numbers \(\alpha_1(r)<1\), \(\beta_1(r)>1\) such that the arc of the level line \(L(f,r)\) is star-shaped for any function \(f(z)\in S\) if and only if
\[ \text{either }\quad \alpha_1(r)\underline{R_1(r)}<|f'(z)|<\overline{R_1(r)}, \]
\[ \text{or}\quad \underline{R_1(r)}<|f'(z)|<\overline{R_1(r)}\beta_1(r), \]
where \(\alpha_1(r)\), \(\beta_1(r)\) are determined from certain equations;
\[ \underline{R_1(r)}=\frac{1-r}{(1+r)^3},\qquad \overline{R_1(r)}=\frac{1+r}{(1-r)^3} \]
are, respectively, the exact lower and upper bounds for \(|f'(z)|\) in the class \(S\).
From this theorem it follows that for any \(r<1\)
\[ \alpha_1(r)\le \alpha_1=\frac{9}{16}e^{\,4\sqrt2\,\arctg\,(1-2\sqrt2)/(1+2\sqrt2)}=0.047\ldots, \]
where
\[ \lim_{r\to1}\alpha_1(r)=\alpha_1. \]
Thus, for any function \(f(z)\in S\), an arc of the level line \(L(f,r)\) for which the inequalities
\[ \alpha_1\underline{R_1(r)}<|f'(z)|<\overline{R_1(r)} \]
hold will be star-shaped for every \(r\), \(0<r<1\).
Correspondingly, \(\beta_1(r)\ge1\), and
\[ \lim \beta_1(r)=1. \]
Moscow Institute
of Steel and Alloys
Received
February 6, 1965
REFERENCES
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- I. A. Aleksandrov, V. I. Popov, Sibirsk. matem. zhurn., 6, 1 (1965).
- O. V. Stepanova, Matem. sborn., 61 (103), 351 (1963).