MATHEMATICS

I. A. ALEKSANDROV

ON THE BOUNDS OF CONVEXITY AND STARLIKENESS FOR FUNCTIONS UNIVALENT AND REGULAR IN A DISK

(Presented by Academician M. A. Lavrentiev on 6 V 1957)

Below we give theorems generalizing the well-known propositions of Nevanlinna \((^1)\) and Grunsky \((^2)\) that any holomorphic univalent function \(z=f(w)\), \(f(0)=0\), \(f'(0)=1\), in the disk \(W:\ |w|<1\) (the totality of such functions is called the class \(S\)) maps the disk with center at \(w=0\) and radius not exceeding \(R_k=2-\sqrt{3}\) onto a convex domain, and the disk with the same center and radius not exceeding \(R_s=\operatorname{th}\frac{\pi}{4}\) onto a domain starlike with respect to \(z=0\), as well as theorems refining and supplementing results obtained by G. M. Goluzin \((^{3,4})\) for bounds of generalized starlikeness.

Theorem 1. The disk with center at the point \(\omega\in W\) and radius \(\rho\leq R_k\),

\[ R_k=2-\sqrt{3+|\omega|^2}, \]

is mapped by any function \(z=f(w)\) of the class \(S\) onto a convex domain. The number \(R_k\) cannot be increased without additional restrictions on \(f(w)\).

The proof is based on the known fact: the range of values of the expression

\[ I=\frac{(w-\omega)f''(w)}{f'(w)}, \]

where \(w\) and \(\omega\) are fixed points of \(W\), and \(f(w)\in S\), is the interior of the disk of radius

\[ \frac{4|w-\omega|}{1-|w|^2} \]

with center at the point

\[ \frac{2(w-\omega)|w|^2}{w(1-|w|^2)}. \]

In the proof of the remaining theorems we use the following lemma, obtained by the variational method.

Lemma. The range of values of the expression

\[ I=\ln \frac{(w-\omega)f'(w)}{f(w)-f(\zeta)} \]

(\(w,\omega,\zeta\in W\) and fixed) in the class \(S\) is the interior of the disk of radius

\[ \ln \frac{|1-\bar{\zeta}w|+|w-\zeta|}{|1-\bar{\zeta}w|-|w-\zeta|}, \]

equal to the non-Euclidean distance between the points \(\zeta\) and \(w\), with center at the point

\[ \ln \frac{(1-|\zeta|^2)(w-\omega)}{(1-w\bar{\zeta})(w-\zeta)}. \]

Theorem 2. Every non-Euclidean disk with non-Euclidean center at the point \(\zeta\in W\), whose non-Euclidean radius is not greater than \(\pi/2\), is mapped by any function of the class \(S\) onto a domain starlike with respect to \(f(\zeta)\). The estimate is sharp.

Following G. M. Goluzin, we shall call a domain of the form \(D_n(a)\) \((n=1,2,\ldots)\) any closed domain all of whose points can be joined to the point \(a\) by a polygonal line lying entirely in it and consisting of no more than \(n\) straight-line segments.

Theorem 3. Every non-Euclidean disk with non-Euclidean center at the point \(\zeta\in W\), whose non-Euclidean radius is not greater than \(n\pi/2\), is mapped by any function \(f(w)\in S\) onto a domain of the form \(D_n(f(\zeta))\). The estimate is sharp.

In particular, every disk of radius not greater than

\[ R_{ns}=\operatorname{th}\frac{n\pi}{4} \]

with center at the point \(w=0\) is mapped by every function of the class \(S\) onto a domain of the form \(D_n(0)\).

Let us note that the proof of Theorems 2 and 3 can be carried out without using the lemma.

Theorem 4. Every disk which contains the point \(w=0\) on its boundary and has radius not greater than \(1/4\) is mapped by any function of the class \(S\) onto a domain starlike with respect to \(z=0\). The indicated bound is sharp.

Theorem 5. Every disk with center at the point \(ae^{i\alpha}\), \(a<1/4\), \(0\leq \alpha\leq 2\pi\), containing the point \(w=0\) in its interior, is mapped by any function of the class \(S\) onto a domain starlike with respect to \(z=0\), if the radius of the disk \(r\) satisfies the inequality

\[ \ln\frac{1-a+r(\sin x-\cos x)}{1-a-r(\sin x+\cos x)} \leq \operatorname{arc\,tg}\frac{r+a\cos x}{a\sin x}, \tag{1} \]

where \(x\), \(0<x<\pi\), is determined from the relation

\[ \sqrt{r^2-2a\cos x+a^2}=\frac{r\sin x}{1-a-r\cos x}. \tag{2} \]

The estimate is sharp.

Denote by \(R(a,0)\) the upper bound of the radii of disks satisfying (1), (2). It can be shown that the value \(R(a,0)\) is not smaller than the value of the larger root of the equation

\[ \ln\frac{1+a+\rho}{1-a-\rho}=\arccos\frac{a}{\rho}. \]

Theorem 6. Every disk with center at the point \(w=0\) is mapped by any function of the class \(S\) onto a domain starlike with respect to the image of the point \(\zeta\) belonging to it, \(|\zeta|=a\), if the radius of the disk \(r\) satisfies the inequality

\[ \ln \frac{ \sqrt{1-2ar\cos x+a^2r^2}+\sqrt{r^2-2ar\cos x+a^2} }{ \sqrt{1-2ar\cos x+a^2r^2}-\sqrt{r^2-2ar\cos x+a^2} } \leq \]

\[ \leq \operatorname{arc\,tg} \frac{r(1+a^2)-a(1+r^2)\cos x}{a(1-r^2)\sin x}, \]

where \(x\), \(0<x<\pi\), is determined from the relation

\[ \frac{a(1-r^2)\,[r(1+a^2)\cos x-a(1+r^2)]} {[r(1+a^2)-a(1+r^2)\cos x]^2+a^2(1-r^2)^2\sin^2 x} = \frac{2ar\sin x} {\sqrt{(1-2ar\cos x+a^2r^2)(r^2-2ar\cos x+a^2)}}. \]

The estimate is sharp.

For the upper bound of the radii \(R(0,a)\) of the disks named in the theorem, one can indicate a sufficiently precise value from below. It is equal to the larger root of the equation

\[ \ln\frac{(1+a)(1+\rho)}{(1-a)(1-\rho)} = \arccos\frac{a(1-\rho^2)}{\rho(1-a^2)}. \]

Theorem 7. Every disk with center at the point \(\zeta=ae^{i\alpha}\in W\), whose radius satisfies the inequality

\[ \ln \frac{(1-a)(\sin x+\cos x)-r}{(1-a)(\sin x-\cos x)+r} \leq \operatorname{arc\,tg}\frac{1-a^2-ar\cos x}{ar\sin x}, \]

where \(x,\ 0<x<\pi\), is determined from the relation

\[ \sqrt{(1-a^2)^2-2ar(1-a^2)\cos x+a^2r^2} = \frac{r(1-a)\sin x}{(1-a)\cos x-r}, \]

is mapped by every function of the class \(S\) onto a domain star-shaped with respect to the image of the center. The estimate is sharp.

The root of the equation

\[ \ln \frac{(1-a)(1+a+\rho)}{(1+a)(1-a-\rho)} = \arccos \frac{a\rho}{1-a^2} \]

gives a sufficiently accurate lower value for the upper bound \(R(a,a)\) of the radii of these disks.

Tomsk State University
named after V. V. Kuibyshev

Received
4 V 1957

CITED LITERATURE

\(^{1}\) R. Nevanlinna, Oversckt av Finska Vet. Soc. Forh. (A), 62, 1 (1919–1920).
\(^{2}\) Grunsky, Jahresber. deutsch. Math. Ver., 43, 140 (1934).
\(^{3}\) G. M. Goluzin, Matem. sborn., 42, No. 2, 169 (1935).
\(^{4}\) G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, 1952.