MATHEMATICS

G. V. KUZ’MINA

SOME COVERING THEOREMS FOR UNIVALENT FUNCTIONS

(Presented by Academician V. I. Smirnov, 3 VIII 1961)

Let (S(a_1,a_2)) be the class of functions (w=f(z)), (f(0)=0), regular and univalent in the disk (|z|<1) and mapping it onto domains containing no points (a_1,a_2); (f(z;a_1,a_2)) is the unique function of this class for which (|f'(0)|\le f'(0;a_1,a_2)), (f(z)\in S(a_1,a_2))*. Let (S) be the class of functions (w=f(z)=z+c_2z^2+\cdots), regular and univalent in the disk (|z|<1); (S_R) is the subclass of functions of the class (S) with real coefficients.

We shall denote, as usual, by (K) the complete elliptic integral of the first kind with modulus (k), and by (\operatorname{sn} z) and (\theta(z)) the Jacobi elliptic functions with the same modulus (k).

Theorem 1. If the function (f(z)=cz+\cdots\in S(a_1,a_2)), (|a_1|=|a_2|=a), (|\arg a_1-\arg a_2|=2\alpha) ((0\le \alpha\le \pi/2)), then the sharp inequality is valid

[
\frac{a}{|c|}\ge h(\alpha)=
\begin{cases}
\dfrac{1}{4}, & \alpha=0,\[6pt]
\left{\sqrt{\dfrac{4pm+(m^2-p^2)^2}{16m^2}\,\dfrac{\theta(0)}{\theta(2u)}}\right}, & 0<\alpha<\dfrac{\pi}{2},\[10pt]
\dfrac{1}{2}, & \alpha=\dfrac{\pi}{2},
\end{cases}
\tag{1}
]

where the functions (u=u(\alpha)), (m=m(\alpha)), (p=p(\alpha)) and (k=k(\alpha)) are uniquely determined for (0<\alpha<\pi/2) by the relations

[
\operatorname{sn} u=m-p\quad (0<u<K),\qquad
\frac{m\cos\alpha-1}{\sqrt{pm}}=\frac{\theta'(u)}{\theta(u)},
]

[
p=\sqrt{m^2-2m\cos\alpha+1},\qquad
k^2=\frac{p+m-\cos\alpha}{2p}.
]

Equality in (1) occurs for (0\le \alpha\le \pi/2) only for the functions (f(z)=f(\varepsilon z;a_1,a_2)), (|\varepsilon|=1). For (0<\alpha<\pi/2) each of the indicated extremal functions maps the disk (|z|<1) onto the entire (w)-plane with a single slit consisting of the ray (\arg w=\arg\sqrt{a_1a_2}), (|w|\ge ma) ((1<m<\infty)), and two circular arcs symmetric with respect to the straight line containing this ray, with ends at the points (a_1,\,m\sqrt{a_1a_2}) and (a_2,\,m\sqrt{a_1a_2}). For (\alpha=0) and (\alpha=\pi/2) the extremal functions

[
\left(f(\varepsilon z;a_1,a_1)=\frac{4a_1\varepsilon z}{(1+\varepsilon z)^2},\quad
f(\varepsilon z;a_1,-a_1)=\frac{2a_1\varepsilon z}{1+\varepsilon^2 z^2},\quad |\varepsilon|=1\right)
]

map the disk (|z|<1) onto the entire (w)-plane with a single radial slit (\arg w=\arg a_1), (|w|\ge a) in the first case, and with two radial slits (\arg w=\arg a_1), (|w|\ge a) and (\arg w=\arg a_2), (|w|\ge a) in the second case**.

The proof of Theorem 1 is based on the use of the known properties of the function (w=f(z)) realizing (\max |f'(0)|) in the given class, and ana-

* The points (a_1,a_2) may be either distinct or coincident.
** Here and below (\arg\sqrt{a_1a_2}) corresponds to the bisector of the smaller angle determined by (\arg a_1,\arg a_2).

... analytic expression for the inverse function (z=f^{-1}(w)), obtained by M. A. Lavrent'ev ((^1)) on the basis of the principles of Lindelöf and Montel, whence there also follows the uniqueness of the extremal function (w=f(z)), normalized by the condition (f'(0)>0). In the limiting cases (\alpha=0) and (\alpha=\pi/2) the assertion of the theorem constitutes the known results of Koebe and Szegő for an unnormalized univalent mapping.

Remark. For (0<\alpha<\pi/2) the analytic expression for the inverse function (z=f^{-1}(w)), where (w=f(z)) is the extremal function of Theorem 1, is determined, up to a rotation, by the quantities (h(\alpha)) and (m(\alpha)) obtained in Theorem 1 and having the geometric meaning indicated there.

Corollary. If the function (f(z)=z+c_2z^2+\cdots \in S) does not assume in the disk (|z|<1) the values (a_1) and (a_2), (|a_1|=|a_2|=a), (|\arg a_1-\arg a_2|=2\alpha) ((0\le \alpha\le \pi/2)), then we have the sharp inequality

[
a\ge h(\alpha)\qquad \left(0\le \alpha\le \frac{\pi}{2}\right),
]

and the equality sign is realized only by the functions (f(z)=f(z;a_1,a_2)).

For (0<\alpha\le \pi/2) this corollary geometrically means that the image of the disk (|z|<1) under the mapping by any function (w=f(z)) of the class (S) contains at least one of the points, equidistant from the origin, of any two segments of length (h(\alpha)), issuing from the point (w=0) at an angle (\alpha) to one another; and the ends of both these segments do not belong to the image of the disk (|z|<1) only for the indicated extremal functions. For (\alpha=0) we obtain the well-known Koebe theorem on covering the disk (|w|<1/4) by the image of the unit disk under its mapping by any function of the class (S).

Denote by (f(|z|<1)) the image of the disk (|z|<1) under the mapping (w=f(z)), and by (\overline{f(|z|<1)}_\beta) the domain symmetric to (f(|z|<1)) with respect to the straight line containing the ray (\arg w=\beta).

From the corollary to Theorem 1 one easily obtains

Theorem 2. The set

[
\bigcap_{f\in S}\left[f(|z|<1)\cup \overline{f(|z|<1)}_\beta\right]
]

of points (w=\rho e^{i\varphi}) is a domain containing the point (w=0), bounded by the curve (w=R_\beta(\varphi)e^{i\varphi}), where (R_\beta(\varphi)=h(|\varphi-\beta|)), (-\pi/2+\beta\le \varphi\le \pi/2+\beta), (R_\beta(\varphi)=h(|\varphi-\beta-\pi|)), (\pi/2+\beta<\varphi<3\pi/2+\beta).

The point (R_\beta(\varphi)e^{i\varphi}) belongs to neither of the domains (f(|z|<1)) and (\overline{f(|z|<1)}\beta) only for the functions
[
f(z)=f\bigl(z;R\beta(\varphi)e^{i\varphi},\,R_\beta(\varphi)e^{i(-\varphi-2\beta)}\bigr)
]
when (|\varphi-\beta|\le \pi/2), and
[
f(z)=f\bigl(z;R_\beta(\varphi)e^{i\varphi},\,R_\beta(\varphi)e^{i(-\varphi+2\beta+2\pi)}\bigr)
]
when (|\varphi-\beta-\pi|<\pi/2).

A consequence of Theorem 2 is

Theorem 3. The set

[
\bigcap_{f\in S_R^*} f(|z|<1)
]

of points (w=\rho e^{i\varphi}) is a domain containing the point (w=0), bounded by the curve (w=R(\varphi)e^{i\varphi}), where (R(\varphi)=h(|\varphi|)), (-\pi/2\le \varphi\le \pi/2), (R(\varphi)=h(|\varphi-\pi|)), (\pi/2<\varphi<3\pi/2).

The point (R(\varphi)e^{i\varphi}) does not belong to the domain (f(|z|<1)) only for the functions
[
f(z)=f\bigl(z;R(\varphi)e^{i\varphi},\,R(\varphi)e^{-i\varphi}\bigr)
]
when (|\varphi|\le \pi/2), and
[
f(z)=f\bigl(z;R(\varphi)e^{i\varphi},\,R(\varphi)e^{i(-\varphi+2\pi)}\bigr)
]
when (|\varphi-\pi|<\pi/2).

Remark. The domain considered in Theorem 3, without a concrete determination of its boundary and of the functions (f(z)) for which the points of this boundary do not belong to the domain (f(|z|<1)), was obtained by Jenkins ((^2)) as one of the examples of the systematic use of the method of the extremal metric and the theory of quadratic differentials.

Let (S^(a_1,a_2)) and (S^) be the sets of those functions of the classes (S(a_1,a_2)) and (S), respectively, which map the disk (|z|<1) onto domains star-shaped with respect to the origin.

From Lindelöf’s principle it easily follows that

Theorem 4. If the function (f(z)=cz+\cdots \in S^*(a_1,a_2)), (|a_1|=|a_2|=a), (|\arg a_1-\arg a_2|=2\pi\lambda) ((0\le \lambda\le 1/2)), then ...

the sharp inequality
[
\frac{a}{|c|}\geq \frac14 \lambda^{-\lambda}(1-\lambda)^{-(1-\lambda)}
\qquad (0\leq \lambda \leq 1/2),
]
and the equality sign is realized only by the functions
[
f(z)=\frac{4\lambda^\lambda(1-\lambda)^{1-\lambda}\sqrt{a_1a_2}\varepsilon z}
{(1-\varepsilon z)^{2\lambda}(1+\varepsilon z)^{2(1-\lambda)}},
\qquad |\varepsilon|=1.
]

For (0<\lambda\leq 1/2) the indicated extremal functions map the disk (|z|<1) onto the entire (w)-plane with two radial slits (\arg w=\arg a_1), (|w|\geq a), and (\arg w=\arg a_2), (|w|\geq a). For (\lambda=0) the extremal functions map the disk (|z|<1) onto the entire (w)-plane with a single radial slit (\arg w=\arg a_1), (|w|\geq a).

Corollary. If the function (f(z)=z+c_2z^2+\ldots\in S^*) does not assume in the disk (|z|<1) the values (a_1) and (a_2), (|a_1|=|a_2|=a), (|\arg a_1-\arg a_2|=2\pi\lambda) ((0\leq \lambda\leq 1/2)), then the sharp inequality
[
|a|\geq \frac14\lambda^{-\lambda}(1-\lambda)^{-(1-\lambda)}
\qquad (0\leq \lambda\leq 1/2),
]
holds, and the equality sign occurs only for the functions
[
f(z)=\frac{z}
{(1-e^{-i\arg\sqrt{a_1a_2}}z)^{2\lambda}
(1+e^{-i\arg\sqrt{a_1a_2}}z)^{2(1-\lambda)}}.
]

This corollary makes it possible to obtain for the class (S^*) certain covering theorems, in the same way as the corollary to Theorem 1 gives the possibility of obtaining covering theorems for the class (S).

REFERENCES

1. M. A. Lavrent'ev, Tr. Fiz.-matem. inst. im. V. A. Steklova AN SSSR, 5, 159 (1934).
2. J. A. Jenkins, Ann. Math., 71, No. 1 (1960).