I. M. MILIN

THE AREA METHOD IN THE THEORY OF UNIVALENT FUNCTIONS

(Presented by Academician M. A. Lavrent’ev on 19 VIII 1963)

We consider the following classes of functions: \(\Sigma\)—the class of functions

\[ F(\zeta)=\zeta+\alpha_0+\alpha_1 \frac{1}{\zeta}+\cdots, \]

regular and univalent in the domain \(1<|\zeta|<\infty\); the classes \(\Sigma_0,\Sigma_2,S,S_2\) are defined as usual; \(\widetilde{\Sigma}, \widetilde{\Sigma}_0, \widetilde{\Sigma}_2\) are subclasses, respectively, of \(\Sigma,\Sigma_0,\Sigma_2\), consisting of functions mapping \(|\zeta|>1\) onto domains without exterior points with boundary of zero area (which is equivalent to the condition \(\sum_{n=1}^{\infty} n|\alpha_n|^2=1\)).

The paper presents a method related to an elementary one, but which makes it possible to obtain, in a unified way, the previously known principal results of the theory of univalent functions and some new results.

I. The system of functions \(\{A_n(z)\}\)

1. Simplest properties of the system \(\{A_n(z)\}\).

Take an arbitrary function \(F(\zeta)\in\Sigma\) and an arbitrary value \(\zeta_1\): \(|\zeta_1|=\rho=1/r>1\). Define in the disk \(|z|<1\) single-valued functions \(A_n(z)\), \(n=1,2,\ldots\), by means of the expansion

\[ \ln \frac{F(\zeta)-F(\zeta_1)}{\zeta-\zeta_1} = \sum_{n=1}^{\infty} A_n(z_1)\frac{1}{\zeta^n}, \qquad z_1=\frac{1}{\zeta_1}, \tag{1} \]

(the branch of the logarithmic function is taken which tends to zero as \(\zeta=\infty\)).

If we take the function \(F_2(\zeta)=\sqrt{F(\zeta^2)}\in\Sigma_2\), when \(F(\zeta)\in\Sigma_0\), and in the same way define the system of functions \(\{a_n(z)\}\):

\[ \ln \frac{F_2(\zeta)-F_2(\zeta_1)}{\zeta-\zeta_1} = \sum_{n=1}^{\infty} a_n(z_1)\frac{1}{\zeta^n}, \qquad z_1=\frac{1}{\zeta_1}, \tag{2} \]

then the systems \(\{A_n(z)\}\) and \(\{a_n(z)\}\) are related by

\[ A_n(z^2)=2a_{2n}(z),\qquad n=1,2,\ldots . \tag{3} \]

The simplest properties of the system \(\{A_n(z)\}\) are easily established:

a) The functions \(A_n(z)\), \(n=1,2,\ldots\), are regular in the disk \(|z|<1\). If the functions \(A_n(z)\), \(n=1,2,\ldots\), are expanded in Taylor series in a neighborhood of \(z=0\),

\[ A_n(z)=\sum_{k=1}^{\infty}\alpha_{nk}z^k,\qquad n=1,2,\ldots, \tag{4} \]

then \(\alpha_{nk}=\alpha_{kn}\), \(n,k=1,2,\ldots\).

b) The system of functions \(\{A_n(z)\}\) is uniformly bounded in the disk \(|z|<1\), i.e., for every \(z\) in the disk \(|z|<1\) and \(n=1,2,\ldots\) we have

\[ |A_n(z)|<C, \tag{5} \]

where \(C\) is an absolute constant.

c) The system \(\{A_n(z)\}\), generated by the function \(w=F(\zeta)\in\Sigma\), is connected with the system of Faber polynomials \(\Phi_n(w)\), generated by the function \(\zeta=\Phi(w)\), inverse to the function \(w=F(\zeta)\), by the equalities

\[ nA_n(z)=\zeta^n-\Phi_n(F(\zeta)),\qquad z=\frac{1}{\zeta},\qquad n=1,2,\ldots . \tag{6} \]

2. Orthogonality of the system \(\{A_n(z)\}\)

Deeper properties of the system \(\{A_n(z)\}\) are obtained with the aid of the following lemma.

Lemma. Let
\[ F(\zeta)=\zeta+a_0+a_1\frac{1}{\zeta}+\cdots \in \Sigma, \]
and let \(Q(w)\) be an arbitrary function, different from a constant, regular in the domain \(g_{\rho_0}\) (\(g_{\rho_0}\) is the interior of the image of the circle \(|\zeta|=\rho_0>1\) under the mapping by the function \(F(\zeta)\)). Suppose that the Laurent expansion of the function \(Q(F(\zeta))\), regular in the annulus \(1<|\zeta|<\rho_0\), has the form

\[ Q(F(\zeta))=\sum_{n=1}^{\infty}\beta_n\frac{1}{\zeta^n}+b_0+\sum_{n=1}^{\infty}b_n\zeta^n,\qquad 1<|\zeta|<\rho_0 . \tag{7} \]

Then

\[ \sum_{n=1}^{\infty} n|\beta_n|^2 \leq \sum_{n=1}^{\infty} n|b_n|^2 . \tag{8} \]

The equality sign occurs if and only if \(F(\zeta)\in\widetilde{\Sigma}\).

This lemma, established earlier by the author \((^1)\) without the conclusion concerning the possibility of the equality sign in (8), is a generalization of the “external area theorem.” Applying it to \(F(\zeta)\in\widetilde{\Sigma}\) and choosing \(Q(w)=\lambda\Phi_n(w)+\Phi_m(w)\), \(n,m=1,2,\ldots\), \(\lambda\) an arbitrary complex number (or \(Q(w)=\lambda\ln(w-w_1)+\ln(w-w_2)\), \(w_1=F(\zeta_1)\), \(w_2=F(\zeta_2)\)), we arrive at the theorem:

Theorem 1. For any system
\[ \left\{A_n(z)=\sum_{k=1}^{\infty} a_{nk}z^k\right\},\qquad n=1,2,\ldots, \]
generated by a function \(F(\zeta)\in\widetilde{\Sigma}\), the system of derivatives \(\{A'_n(z)\}\) is orthogonal in the domain \(|z|<1\). Moreover, the equalities hold

\[ \frac{1}{\pi}\iint_{|z|<1} A'_n(z)\,\overline{A'_m(z)}\,d\sigma = \sum_{k=1}^{\infty} k a_{nk}\overline{a_{mk}} = \begin{cases} 0, & \text{if } m\ne n,\\[2mm] \dfrac{1}{n}, & \text{if } m=n. \end{cases} \tag{9} \]

It follows from Theorem 1 that the system of functions \(\{\sqrt{n}\,A_n(z)\}\), generated by a function \(F(\zeta)\in\widetilde{\Sigma}\), is orthonormal in the domain \(|z|<1\). It is not difficult to prove the completeness of this system in the class of functions regular in the unit disk and with integrable square of the modulus in \(|z|<1\). This makes it possible to formulate a theorem on the expandability of regular functions in the system \(\{A_n(z)\}\).

Theorem 2. If the system \(\{A_n(z)\}\) is generated by a function \(F(\zeta)\in\widetilde{\Sigma}\), then for any function
\[ f(z)=\sum_{n=1}^{\infty} c_n z^n \]
regular in the disk \(|z|<1\) and with finite area of the image of this disk, there is an expansion in the system \(\{A_n(z)\}\) (10) and the area formula (11):

\[ f(z)=\sum_{n=1}^{\infty}\lambda_n A_n(z),\qquad \lambda_n=\frac{n}{\pi}\iint_{|z|<1} f'(z)\,\overline{A'_n(z)}\,d\sigma; \tag{10} \]

\[ \sum_{n=1}^{\infty} n|c_n|^2=\sum_{n=1}^{\infty}\frac{|\lambda_n|^2}{n}. \tag{11} \]

Moreover, the series \(\sum_{n=1}^{\infty}\lambda_n A_n(z)\) converges absolutely and uniformly inside the circle \(|z|<1\).

Finally, using the orthonormality of the system \(\{\sqrt{n}\,A_n(z)\}\), we obtain the sharp estimates (12) and (13).

Theorem 3. For any systems \(\{A_n(z)\}\) and \(\{a_n(z)\}\), \(n=1,2,\ldots\), generated respectively by the functions \(F(\zeta)\in\Sigma\) and \(F_2(\zeta)\in\Sigma_2\), the following inequalities hold in the circle \(|z|<1\):

\[ \sum_{n=1}^{\infty} n|A_n(z)|^2 \le \ln \frac{1}{1-r^2}, \qquad r=|z|; \tag{12} \]

\[ \sum_{n=1}^{\infty} (2n-1)|a_{2n-1}(z)|^2 \le \frac12 \ln \frac{1+r^2}{1-r^2}, \qquad \sum_{n=1}^{\infty} 2n|a_{2n}(z)|^2 \le \frac12 \ln \frac{1}{1-r^4}, \qquad r=|z|. \tag{13} \]

Equality in (12) holds if and only if \(F(\zeta)\in\widetilde{\Sigma}\), and in (13) if \(F_2(\zeta)\in\widetilde{\Sigma}_2\).

II. Univalent functions

We indicate some applications of the remarkable properties of the system \(\{A_n(z)\}\).

1. Distortion theorems. From the definition of the systems \(\{A_n(z)\}\) and \(\{a_n(z)\}\) we obtain the following expressions for certain functionals in the classes \(\Sigma_0\) and \(S\):

\[ \ln F'(\zeta)=\sum_{n=1}^{\infty} A_n(z) z^n; \]

\[ \ln \frac{F_2(\zeta)}{\zeta} = \ln \frac{z}{f_2(z)} = \sum_{n=1}^{\infty}(-1)^n a_n(z) z^n, \qquad z=\frac1{\zeta}; \tag{14} \]

\[ \ln \frac{\zeta F_2'(\zeta)}{F_2(\zeta)} = \ln \frac{z f_2'(z)}{f_2(z)} = 2\sum_{n=1}^{\infty} a_{2n-1}(z) z^{2n-1}. \]

Estimating the modulus of each of the sums in (14) by means of Cauchy’s inequality and taking into account the estimates (12) and (13), we obtain the classical distortion theorems. The Goluzin “distortion of chords” theorems \((^2)\) and Bazilevich’s “distortion of central angles” theorems \((^3)\) are proved analogously. Thus Theorem 3 is a universal distortion theorem in the classes \(S\) and \(\Sigma\).

From the orthonormality of the system \(\{\sqrt{n}\,A_n(z)\}\) there also easily follow the generalized distortion-type theorems obtained by G. M. Goluzin.

2. Conditions for univalence. Let the function \(F(\zeta)\) be given in the domain \(|\zeta|>1\) by the formal Laurent series

\[ F(\zeta)=\zeta+a_0+\sum_{n=1}^{\infty} a_n \zeta^{-n}. \tag{15} \]

In the exterior of the unit circle form the formal expansion of the function
\(\ln \dfrac{F(\zeta)-F(t)}{\zeta-t}\) in a double power series:

\[ \ln \frac{F(\zeta)-F(t)}{\zeta-t} = \sum_{n,k=1}^{\infty} \alpha_{nk}\zeta^{-n}t^{-k}. \tag{16} \]

where \(a_{nk}\) are integral rational functions of \(a_1, a_2, \ldots, a_m\), \(m=\max(n,k)\). Then Theorem 1 is not difficult to reformulate in the form of conditions for the univalence of \(F(\zeta)\).

Theorem 4. In order that the function \(F(\zeta)\), given by formula (15), be regular and univalent in the domain \(1<|\zeta|<\infty\), it is necessary and sufficient that the coefficients \(a_{nk}\) of the formal expansion (16) satisfy the system of inequalities:
\[ \sum_{k=1}^{\infty} k |a_{nk}|^2 \leqslant \frac{1}{n},\qquad n=1,2,\ldots \tag{17} \]

Moreover, if in (17) the equality sign holds for at least one value of \(n\), then \(F(\zeta)\in \widetilde{\Sigma}\) (and the equality sign in (17) will hold for every \(n\geqslant 1\)).

For the coefficients \(a_{nk}\) \((n,k=1,2,\ldots)\), generated by functions \(F(\zeta)\in\widetilde{\Sigma}\), for arbitrary values of the complex variables \(x_n\) \((n=1,2,\ldots,m)\), \(m\geqslant 1\), from Theorem 2 we obtain the identity
\[ \sum_{n=1}^{\infty} n\left|\sum_{k=1}^{m} x_k a_{nk}\right|^2 = \sum_{n=1}^{m}\frac{|x_n|^2}{n}. \tag{18} \]

Identity (18) immediately leads to the estimate
\[ \left|\sum_{n,k=1}^{m} a_{nk}x_nx_k\right| \leqslant \sum_{n=1}^{m}\frac{|x_n|^2}{n}, \tag{19} \]
obtained by M. Schiffer\(^4\) by the method of variations and by G. M. Goluzin\(^5\) by Löwner’s parametric method.

3. Mean integral modulus.

Theorem 5. For the function \(f(z)=\overline{f(\overline z)}\in S\), for any \(r\), \(0<r<1\), we have the sharp estimate
\[ \frac{1}{2\pi}\int_{|z|=r} |f(z)|\,d\arg z \leqslant \frac{r}{1-r^2}. \tag{20} \]

The equality sign occurs only for the functions
\[ f(z)=\frac{z}{(1\pm z)^2}. \]

Received
19 VIII 1963

References Cited

\(^1\) N. A. Lebedev, I. M. Milin, Matem. sborn., 28 (70), 2, 360 (1951).
\(^2\) G. M. Goluzin, Matem. sborn., 19 (61), 2, 183 (1946).
\(^3\) I. E. Bazilevich, Matem. sborn., 28 (70), 1, 147 (1951).
\(^4\) M. Schiffer, Bull. Am. Math. Soc., 54, 503 (1948).
\(^5\) G. M. Goluzin, Matem. sborn., 29 (71), 1, 197 (1951).