MATHEMATICS

G. V. KUZ’MINA

COVERING THEOREMS FOR FUNCTIONS REGULAR AND UNIVALENT IN THE DISK

(Presented by Academician V. I. Smirnov on 22 VI 1964)

Let \(S(a_1,a_2)\) be the class of functions \(f(z)=c_1z+\cdots\), regular and univalent in the disk \(B:\ |z|<1\) and not assuming in it the prescribed values \(a_1\) and \(a_2\); \(S^*(a_1,a_2)\) the subclass of starlike functions from \(S(a_1,a_2)\); \(f(z;a_1,a_2)\) that function of the class \(S(a_1,a_2)\) for which \(|f'(0)|\le f'(0;a_1,a_2)\), \(f(z)\in S(a_1,a_2)\); \(f^*(z;a_1,a_2)\) the analogous extremal function for the class \(S^*(a_1,a_2)\).

Let \(S\) be the class of functions \(f(z)=z+c_1z^2+\cdots\), regular and univalent in \(B\); \(S^*\) the subclass of starlike functions from \(S\); \(S_+(S_+^*)\) the set consisting of the function \(f(z)=z\) and all those functions of the class \(S(S^*)\) for which, among the coefficients \(c_2,c_3,\ldots\), the first coefficient different from zero is positive.

In this paper \(\max |f'(0)|\) is found in the classes \(S(a_1,a_2)\) and \(S^*(a_1,a_2)\), and the functions \(f(z;a_1,a_2)\) and \(f^*(z;a_1,a_2)\) are determined. With the aid of these results some covering theorems are established in the classes \(S\), \(S^*\), and the largest sets belonging to the image of the disk \(B\) under its mapping by any function of the classes \(S_+\) and \(S_+^*\), respectively, are determined.

In what follows, \(K(k)\) is the complete elliptic integral of the first kind with modulus \(k\); \(\operatorname{sn}u\), \(\operatorname{cn}u\), \(\operatorname{dn}u\), \(\theta(u)\) are Jacobi elliptic functions with the same modulus \(k\); \(K'(k)=K(\sqrt{1-k^2})\).

Let \(0<a\le1\), \(0\le\alpha\le\pi/2\); \(m,p,\omega,k,\delta\) be continuous functions of \(a\) and \(\alpha\), defined for \(0<a\le1\), \(0<\alpha<\pi/2\) by the system

\[ p=\sqrt{m^2-\left(e^{-i\alpha}+\frac{1}{a^2}e^{i\alpha}\right)m+\frac{1}{a^2}}\;{}^*,\qquad k^2=\frac{p+m-\frac12\left(e^{-i\alpha}+\frac{1}{a^2}e^{i\alpha}\right)}{2p}, \]

\[ \operatorname{cn}\omega=\frac{m-p}{m+p},\qquad \left(\frac{\sqrt{p/m}}{a(m+p)}-\frac{\theta'(\omega)}{\theta(\omega)}\right)K=i\delta,\qquad 2\delta\,\operatorname{Im}(iK'/K)=\pi\,\operatorname{Im}(\omega/K) \tag{1} \]

\[ (\omega=\lambda_1K+\lambda_2 iK',\qquad \text{where }0<\lambda_1<1,\quad -1<\lambda_2<1); \]

\[ h(a,\alpha)=\frac{a^2}{4}\left|\frac{(m+p)^2\theta^2(0)}{m\theta^2(\omega)}\right|e^{\delta\,\operatorname{Im}\omega/K},\qquad 0<\alpha<\frac{\pi}{2}; \]

\[ h(a,0)=\frac14,\qquad h\left(a,\frac{\pi}{2}\right)=\frac{1+a^2}{4}; \]

\(w=F(z,a,\alpha)\) is determined for \(z\in B\), \(0<a\le1\), by the conditions

\[ w=m-p\,\frac{1+\operatorname{cn}u}{1-\operatorname{cn}u},\qquad ze^{-i\psi}= \frac{ \operatorname{sn}\frac{u-\omega}{2}\, \operatorname{dn}\frac{u-\omega}{2}\, \theta^2\!\left(\frac{u-\omega}{2}\right) }{ \operatorname{sn}\frac{u+\omega}{2}\, \operatorname{dn}\frac{u+\omega}{2}\, \theta^2\!\left(\frac{u+\omega}{2}\right) } e^{-i\delta u/K}\;{}^{**}. \]

\[ \psi=\arg\left\{\frac{m\theta^2(\omega)}{(m+p)^2\theta^2(0)}e^{i\delta\omega/K}\right\},\qquad 0<\alpha<\frac{\pi}{2}; \]

\[ F(z,a,0)=\frac{4z}{(1+z)^2},\qquad F\left(z,a,\frac{\pi}{2}\right)=\frac{4}{1+a^2}\, \frac{z}{1+2i\frac{1-a^2}{1+a^2}z-z^2}. \]

\[ \text{* Here and below that branch of the square root is considered whose values are positive for positive values of the radicand.} \]

\[ \text{** By the function }u=u(z,a,\alpha)\text{ is meant that branch for which }u(0,a,\alpha)=\omega(a,\alpha). \]

Theorem 1. In the class \(S\left(e^{-i\alpha}, \dfrac{1}{a^2}e^{i\alpha}\right)\), \(0<a\leqslant 1\), \(0\leqslant \alpha \leqslant \pi/2\), of functions \(f(z)=c_1z+\ldots\), the sharp inequality

\[ |c_1|\leqslant \frac{1}{h(a,\alpha)}. \tag{2} \]

holds.

Equality in (2) occurs only for the function
\[ f\left(\varepsilon z; e^{-i\alpha}, \frac{1}{a^2}e^{i\alpha}\right), \quad |\varepsilon|=1, \]
where
\[ f\left(z; e^{-i\alpha}, \frac{1}{a^2}e^{i\alpha}\right)=F(z,a,\alpha). \]
For \(0<\alpha<\dfrac{\pi}{2}\), each of the extremal functions maps the disk \(B\) onto the whole \(w\)-plane with a single slit consisting of three analytic arcs, two of which join, respectively, the points \(e^{-i\alpha}\), \(m\) and \(\dfrac{1}{a^2}e^{i\alpha}\), \(m\), while the third has its origin at the point \(m\) and goes to infinity. For \(\alpha=0\) and \(\alpha=\pi/2\), the extremal functions map the disk \(B\) onto the whole \(w\)-plane with a single radial slit \(w\geqslant 1\) in the first case, and with two radial slits \(w=-it\), \(t\geqslant 1\), and \(w=it\), \(t\geqslant 1/a^2\), in the second case.

The proof of the theorem uses known properties of the function \(w=f(z)\) realizing \(\max |f'(0)|\) in the given class, in particular the uniqueness of the extremal function normalized by the condition \(f'(0)>0\), and the differential equation for the extremal function (1). In the limiting cases \(\alpha=0\) and \(\alpha=\pi/2\), we obtain the known results of Koebe and M. A. Lavrent'ev for the mapping considered in the theorem.

Using the simple relation between the class \(S(a_1,a_2)\), where \(a_1,a_2\) are arbitrary prescribed values, and the class
\[ S\left(e^{-i\alpha}, \frac{1}{a^2}e^{i\alpha}\right) \]
for suitable \(a\) and \(\alpha\), from Theorem 1 we immediately obtain

Corollary 1. In the class \(S(a_1,a_2)\), \(0<|a_1|\leqslant |a_2|\), of functions \(f(z)=c_1z+\ldots\), the inequality holds \(\bigl(a^2=|a_1/a_2|,\ \alpha=\arg\sqrt{a_1/a_2},\ -\pi/2<\alpha\leqslant \pi/2\bigr)\)

\[ |c_1|\leqslant \frac{|a_1|}{h(a,|\alpha|)}. \tag{3} \]

Equality in (3) is realized only by the function \(f(\varepsilon z;a_1,a_2)\),
\[ |\varepsilon|=1,\quad f(z;a_1,a_2)=a_1e^{i\alpha} f\left(\frac{|a_1|}{a_1}e^{-i\alpha}z;\ e^{-i\alpha},\ \frac{1}{a^2}e^{i\alpha}\right) \]
\[ \bigl(f(z;\overline{a_1},\overline{a_2})=\overline{f}(z;a_1,a_2)\bigr)^*. \]

Corollary 2. Let \(a\), \(0<a\leqslant 1\), and \(\alpha\), \(-\pi/2<\alpha\leqslant \pi/2\), be prescribed numbers, and let \(f(z)\in S\) not take in the disk \(B\) the values \(a_1\) and \(a_2\), for which \(a_1/a_2=a^2e^{2i\alpha}\). Then the inequality holds

\[ |a_1|\geqslant h(a,|\alpha|). \tag{4} \]

Equality in (4) occurs only for functions
\[ f\left(z; a_1, \frac{1}{a^2}e^{-i\alpha}a_1\right), \]
where \(a_1=h(a,\alpha)e^{i\beta}\), and \(\beta\) is an arbitrary real number.

For \(a=1\), from Theorem 1 and its corollaries we obtain, as a special case, the results stated in \((^2)\).

Remark. From the differential equation for the function
\[ f\left(z; e^{-i\alpha}, \frac{1}{a^2}e^{i\alpha}\right) = c_1(a,\alpha)z+c_2(a,\alpha)z^2+\ldots \]
\((^1)\), and from the system (1), we find that, for \(0<a\leqslant 1\), \(0\leqslant |\alpha|<\pi/2\), there is defined a continuous function

\[ \gamma(a,\alpha)=\arg c_2(a,\alpha),\qquad \gamma(1,\alpha)=\pi \quad\text{for}\quad 0\leqslant |\alpha|<\pi/2. \]

Let \(f(B)\) be the image of the disk \(B\) in the \(w\)-plane under the mapping \(w=f(z)\); let \(\overline{f}(B)\) be the domain symmetric to \(f(B)\) with respect to the real axis.

* The function \(\overline{f}(z)\) is obtained from the function \(f(z)=c_1z+\ldots+c_nz^n+\ldots\) by replacing all its coefficients \(c_n\) by \(\overline{c}_n\).

Set
\[ D(S)=\bigcap_{f\in S} f(B),\qquad \mathfrak D(S)=\bigcap_{f\in S}\bigl[f(B)\cup \overline{f(B)}\bigr]; \]
the corresponding sets for subclasses of \(S\) are denoted analogously.

From the results obtained from Theorem 1 and its corollaries for \(a=1\), and presented in \((2)\), one easily finds the set \(\mathfrak D(S)\), which coincides with the largest set belonging to the image of the disk \(B\) under its mapping by any function of the class \(S\) with real coefficients, and all its boundary functions \((2)\) are determined.

It is well known (and follows in an obvious way from Corollary 2 to Theorem 1) that the set \(D(S)\) is the disk \(|w|<1/4\), and to each boundary point \(w_\theta=\frac14 e^{i\theta}\) of it there corresponds only the function
\[ f_\theta(z)=\frac{z}{(1+e^{-i\theta}z)^2}. \]
To obtain a strengthening of this result of Koebe, it is natural to consider the class \(S_+\) \((3)\). Clearly, the set \(D(S_+)\) is symmetric with respect to the real axis, and it follows from simple considerations that the part of the boundary \(w=R_+(\varphi)e^{i\varphi}\) (\(R_+,\varphi\) are polar coordinates) of this set lying in the half-plane \(\operatorname{Re} w\ge 0\) is the semicircle \(w=\frac12 e^{i\varphi}\), \(|\varphi|\le \pi/2\) (with all interior points of this semicircle belonging to the set under consideration), \(R_+(\pi)=1/4\) \((3)\). With the aid of Theorem 1 and the results following from it, the function \(R_+(\varphi)\) is determined in principle for \(0<|\varphi-\pi|<\pi/2\). This gives the set \(D(S_+)\).

Let \(\varphi_0\) be fixed, \(0<\varphi_0<\pi/2\); let \(\rho(\varphi_0)\) be the least value of the function \(h(a,|\alpha|)\), \(0<a\le 1,\ 0<|\alpha|<\pi/2\), under the condition
\[ \gamma(a,\alpha)-\alpha=\pi-\varphi_0. \]

Theorem 2. The set \(D(S_+)\) is bounded by the curve \(w=R_+(\varphi)e^{i\varphi}\), where
\[ R_+(\varphi)=\frac12,\quad |\varphi|\le \pi/2;\qquad R_+(\varphi)=\rho(|\pi-\varphi|),\quad 0<|\pi-\varphi|<\pi/2; \]
\[ R_+(\pi)=\frac14. \]

The semicircle \(w=\frac12 e^{i\varphi}\), \(-\pi/2<\varphi<\pi/2\), belongs to the set \(D(S_+)\); the remaining boundary points of the set \(D(S_+)\) do not belong to it.

We now give the solution of the same problems for starlike functions.

Let, for \(0<a\le 1,\ 0\le \lambda\le 1\), \(\psi=\psi(a,\lambda)\) \((0\le \psi\le \pi/2)\) be the solution of the equation
\[ g(\psi,\lambda)= \left( \frac{1+\lambda\cos^2\psi-\sin\psi\sqrt{1-\lambda^2\cos^2\psi}} {1+\lambda\cos^2\psi+\sin\psi\sqrt{1-\lambda^2\cos^2\psi}} \right)^{(1+\lambda)/2} \times \]
\[ \times \left( \frac{1-\lambda\cos^2\psi-\sin\psi\sqrt{1-\lambda^2\cos^2\psi}} {1-\lambda\cos^2\psi+\sin\psi\sqrt{1-\lambda^2\cos^2\psi}} \right)^{(1-\lambda)/2} =a^2. \]

Let \(0<a\le 1,\ 0\le \alpha\le \dfrac{\pi}{2}\); \(\lambda=1-\dfrac{2}{\pi}\alpha\);
\[ h^*(a,\alpha)=\frac12 \left(1+\lambda\cos^2\psi+\sin\psi\sqrt{1-\lambda^2\cos^2\psi}\right)^{-(1+\lambda)/2}\times \]
\[ \times \left(1-\lambda\cos^2\psi+\sin\psi\sqrt{1-\lambda^2\cos^2\psi}\right)^{-(1-\lambda)/2}, \qquad 0<\alpha<\frac{\pi}{2}; \]
\[ h^*(a,0)=\frac14,\qquad h^*\left(a,\frac{\pi}{2}\right)=\frac{1+a^2}{4}; \]
\[ F^*(z,a,\alpha)= \frac{1}{h^*(a,\alpha)} \frac{z}{(1+e^{i(1-\lambda)\psi}z)^{1+\lambda}(1-e^{-i(1+\lambda)\psi})^{1-\lambda}}, \qquad |z|<1^*. \]

The following is based directly on Lindelöf’s principle.

Theorem 3. In the class \(S^*(a_1,a_2)\), \(0<|a_1|\le |a_2|\), of functions \(f(z)=c_1z+\ldots\), the inequality
\[ \left( a^2=\frac{a_1}{a^2},\quad d=\arg\sqrt{\frac{a_1}{a^2}},\quad -\frac{\pi}{2}<\alpha\le \frac{\pi}{2} \right) \qquad |c_1|\le \frac{|a_1|}{h^*(a,|\alpha|)}. \tag{5} \]

\[ \underline{\phantom{xxxxxxxx}} \]

* Here and below, by the function \(F^*(z,a,\alpha)\) is meant that branch of it for which
\[ \frac{h^*(a,\alpha)F^*(z,a,\alpha)}{z}\to 1 \quad\text{as } z\to 0. \]

The equality sign in (5) occurs only for the functions \(f^*(\varepsilon z;\, a_1, a_2)\), \(|\varepsilon|=1\),

\[ f^*(z;\, a_1, a_2)=a_1 e^{i\alpha}F^*\left(\frac{|a_1|}{a_1}e^{-i\alpha}z,\, a,\, \alpha\right) \quad \text{for } 0\le \alpha\le \pi/2, \]

\[ f^*(z;\, \bar a_1, \bar a_2)=\overline{f^*(z;\, a_1, a_2)}. \]

For \(0<|\alpha|<\pi/2\), each of the extremal functions maps the disk \(B\) onto the whole \(w\)-plane with two radial slits \(\arg w=\arg a_1\), \(|w|\ge |a_1|\), and \(\arg w=\arg a_2\), \(|w|\ge |a_2|\); for \(\alpha=0\), onto the whole \(w\)-plane with a single radial slit \(\arg w=\arg a_1\), \(|w|\ge |a_1|\).

Let, for \(0<\varphi_0<\pi/2\), \(\lambda_0=\lambda(\varphi_0)\), \(\chi_0=\chi(\varphi_0)\) \((0<\lambda_0<1,\; 0<\chi_0<\pi/2)\) be the solution of the system

\[ \ln \frac{1+\lambda}{\sqrt{1-\lambda^2\sin^2\chi}+\lambda\cos\chi} - \frac{\sin\chi}{\sqrt{1-\lambda^2\sin^2\chi}+\cos\chi}\, \frac{\chi(1+\lambda^2\tg^2\chi)-\tg\chi}{\lambda\tg^2\chi} =0, \]

\[ \lambda\tg\chi=\tg(\varphi_0+\lambda\chi); \]

\[ \rho^*(\varphi_0) = \frac12 \left(1+\lambda_0\sin^2\chi_0+\cos\chi_0\sqrt{1-\lambda^2\sin^2\chi_0}\right)^{-(1+\lambda_0)/2} \left(1-\lambda_0\sin^2\chi_0+\cos\chi_0\sqrt{1-\lambda_0^2\sin^2\chi_0}\right)^{-(1-\lambda_0)/2}. \]

Theorem 4. The set \(D(S_+^*)\) is bounded by the curve \(w=R_+^*(\varphi)e^{i\varphi}\), where \(R_+^*(\varphi)=1/2\), \(|\varphi|\le \pi/2\); \(R_+^*(\varphi)=\rho^*(|\pi-\varphi|)\), \(0<|\pi-\varphi|<\pi/2\); \(R_+^*(\pi)=1/4\).

The semicircle \(w=\frac12 e^{i\varphi}\), \(-\pi/2<\varphi<\pi/2\), belongs to the set \(D(S_+^*)\); the remaining boundary points of this set do not belong to it. The point \(w_\pi=-1/4\) does not belong to the domain \(f(B)\), where \(f\in S_+^*\), only for the function \(f(z)=z/(1-z)^2\); the points \(w_{-\pi/2}=-i/2\) and \(w_{\pi/2}=i/2\) only for the function \(f(z)=z/(1-z^2)\); the point \(w_\varphi=R_+^*(\varphi)e^{i\varphi}\) for \(\pi/2<\varphi<\pi\) only for the function

\[ f^*\left(z;\, R_+^* e^{i\varphi},\, \frac{1}{\mu^2}R_+^* e^{i[\varphi+(1-\lambda_0)\pi]}\right), \]

for \(\pi<\varphi<3\pi/2\) only for the function

\[ f^*\left(z;\, R_+^* e^{i\varphi},\, \frac{1}{\mu^2}R_+^* e^{i[\varphi-(1-\lambda_0)\pi]}\right), \]

where \(\mu=g(\pi/2-\chi_0,\lambda_0)\), \(\lambda_0=\lambda_0(|\pi-\varphi|)\), \(\chi_0=\chi_0(|\pi-\varphi|)\).

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
13 VI 1964

REFERENCES

1. M. A. Lavrent'ev, Tr. Fiz.-matem. inst. im. V. A. Steklova AN SSSR, 5, 159 (1934).
2. G. V. Kuz'mina, DAN, 142, No. 1, 29 (1962).
3. W. T. Scott, Am. Math. Monthly, 64, No. 8, pt. 2, 90 (1957).