V. S. ZAKHARYAN

ON A UNIQUENESS THEOREM

(Presented by Academician V. A. Ambartsumian, 18 XI 1963)

1°. As usual, by \(A\) we denote the class of functions \(w(\xi)\) regular in \(|\xi|<1\) for which the integral

\[ \int_0^{2\pi} \lg^+ |w(re^{i\vartheta})|\,d\vartheta \]

is bounded as \(r \to 1-0\). It is known that functions of the class \(A\) have finite boundary values almost everywhere along nontangential paths, which are called their boundary values. If \(w(\xi)\in A\) has zero boundary values on a set \(E\) of positive measure on \(|\xi|=1\), then \(w(\xi)\equiv 0\) in \(|\xi|<1\) (see, for example, \((^1)\)).

Let us pose the following question: is it possible, by narrowing the class \(A\), to achieve that an analogous situation holds on more sparse sets? Below a positive answer to this question is given.

2°. For functions \(w(z)\) meromorphic in the disk \(|z|<1\), denote

\[ T(r)=\int_0^r \frac{A(t)}{t}\,dt, \]

where \(T(r)\) is the characteristic function in the sense of Nevanlinna, and \(A(r)\) is the area of the domain of the Riemann surface onto which the function \(w(z)\) maps the disk \(|z|<r\). \(A(r)\) is measured in the spherical metric.

In the work \((^2)\), M. M. Dzhrbashyan considered the classes \(A(\alpha)\) of functions meromorphic in the unit disk whose characteristics satisfy the condition

\[ \int_0^1 (1-r)^\alpha T(r)\,dr<\infty \qquad (\alpha>-1), \]

and obtained an integral representation of functions of these classes.

In particular, if a holomorphic function \(w(z)\) from \(A(\alpha)\) has no zeros, then it is represented in the form

\[ w(z)=c_0\exp\left[ \frac{1}{\pi}\int_0^1\int_0^{2\pi} (1-\rho^2)^\alpha \frac{\lg |w(\rho e^{i\vartheta})|} {(1-z\rho e^{-i\vartheta})^{\alpha+2}} \,\rho\,d\rho\,d\vartheta \right]. \tag{1} \]

Following L. Carleson \((^3)\), we introduce the following class of functions. We shall say that a function meromorphic in the disk \(|z|<1\) belongs to the class \(T_\alpha\), \(0<\alpha\le 1\), if

\[ T_\alpha(w)\equiv \int_0^1 \frac{A(r)}{(1-r)^\alpha}\,dr<\infty \qquad \text{for } 0<\alpha<1, \]

\[ T_1(w)\equiv \lim_{r\to 1-0} A(r)<\infty \qquad \text{for } \alpha=1. \]

We note that since \(T_\alpha \subset A(\alpha)\), for functions of the class \(T_\alpha\) having no zeros, the representation (1) is valid.

For functions of the classes \(T_\alpha\), L. Carleson proved \((^3)\) the following theorems.

A. \(\displaystyle \lim_{r\to 1-0} w(re^{i\vartheta})\) exists for all \(\vartheta\), except, possibly, for a set whose outer \((1-\alpha)\)-capacity is zero.

B. If \(w(z)\not\equiv a\), then the outer \((1-\alpha)\)-capacity of the set of values \(\vartheta\) satisfying the equality \(\displaystyle \lim_{r\to 1-0} w(re^{i\vartheta})=a\) is zero, with the possible exception of values \(a\) belonging to a set of plane measure zero.

Since assertion B, generally speaking, does not hold for all \(a\), it is impossible to obtain from it a uniqueness theorem in the classical form.

\(3^\circ\). We shall prove the following theorem.

Theorem 1. Let a function \(w(z)\), holomorphic in \(|z|<1\), belong to the class \(T_\alpha\), \(0<\alpha\leq 1\), and let \(w(a_\nu)=0\). If

\[ \sum (1-|a_\nu|)^{1-\alpha}<+\infty \]

and the limiting radial values of the function \(w(z)\) are equal to zero on a set \(E\), whose outer \((1-\alpha)\)-capacity is positive, then \(w(z)\equiv 0\) in \(|z|<1\).

Proof. Since \(w(z)\in T_\alpha\), it can be represented in the form

\[ w(z)=B(z)w_1(z), \tag{2} \]

where \(B(z)\) is a Blaschke product, and the function \(w_1(z)\) belongs to the class \(T_\alpha\) \((^3)\).

Let \(E\subset[0,2\pi]\) be a certain set whose \((1-\alpha)\)-capacity is positive. Then there is a unit distribution \(\mu\) on \(E\) such that

\[ \int_0^{2\pi}\frac{d\mu(t)}{|e^{it}-re^{ix}|^{1-\alpha}}<c_1 \tag{3} \]

uniformly for all \(x\in[0,2\pi]\) and \(r<1\) \((^3)\).

We shall prove that

\[ \int_0^{2\pi}\left|\lg|w(re^{i\vartheta})|\right|\,d\mu(\vartheta)<c_2, \tag{4} \]

where \(c_2\) does not depend on \(r\).

Denote by

\[ B^{(r)}(z)=\prod_{|a_\nu|<r}\frac{a_\nu-z}{1-\overline{a}_\nu z}\cdot\frac{\overline{a}_\nu}{|a_\nu|}; \]

then, if \(z=re^{i\vartheta}\), we have

\[ \int_0^{2\pi}\left|\lg|B^{(r)}(z)|\right|\,d\mu(\vartheta) \leq \sum_{|a_\nu|<r}\int_0^{2\pi} \left|\lg\left|\frac{a_\nu-z}{1-\overline{z}a_\nu}\cdot\frac{\overline{a}_\nu}{|a_\nu|}\right|\right| \,d\mu(\vartheta) \leq \]

\[ \leq c_3\sum_{|a_\nu|<r}(1-|a_\nu|) \int_0^{2\pi}\frac{d\mu(\vartheta)}{|z-a_\nu|} \leq c_3\sum_{|a_\nu|<r}\frac{1-|a_\nu|}{(r-|a_\nu|)^\alpha} \int_0^{2\pi}\frac{d\mu(\vartheta)} {|re^{i\vartheta}-|a_\nu|e^{i\vartheta_\nu}|^{1-\alpha}}. \]

Hence it is easy to see that

\[ \int_0^{2\pi}\left|\lg|B(z)|\right|\,d\mu(\vartheta) \leq c_4\sum_1^\infty(1-|a_\nu|)^{1-\alpha} \int_0^{2\pi} \frac{d\mu(\vartheta)} {|1-|a_\nu|e^{i(\vartheta-\vartheta_\nu)}|^{1-\alpha}} <c_5. \tag{5} \]

Now we use the integral representation (1) of the function \(w_1(z)\). We have

\[ \int_0^{2\pi}\left|\lg|w_1(re^{i\varphi})|\right|\,d\mu(\varphi) \leq \]

\[ \leq c_6+\frac{1}{\pi}\int_0^1(1-\rho^2)^\alpha\rho\,d\rho \int_0^{2\pi}\left|\lg|w_1(\rho e^{i\vartheta})|\right|\,d\vartheta \int_0^{2\pi}\operatorname{Re}\frac{1}{(1-z\rho e^{-i\vartheta})^{\alpha+2}}\,d\mu(\varphi) \tag{6} \]

or

\[ \int_0^{2\pi}\left|\lg|w_1(re^{i\varphi})|\right|\,d\mu(\varphi) \leq c_6+c_7\int_0^1(1-\rho^2)^\alpha\rho T_1(\rho)\,d\rho \int_0^{2\pi}\operatorname{Re}\frac{1}{(1-z\rho e^{-i\vartheta})^{\alpha+2}}\,d\mu(\varphi). \]

Changing the order of integration and integrating by parts, we obtain

\[ \int_{0}^{2\pi} \left|\lg |w_1(re^{i\varphi})|\right|\,d\mu(\varphi) \le \]

\[ \le c_6+c_8\int_{0}^{1}(1-\rho^2)^{\alpha+1}A_1(\rho)\,d\rho \int_{0}^{2\pi} \frac{d\mu(\varphi)} {\left|1-r\rho e^{i(\varphi-\theta)}\right|^{1-\alpha} \left|1-r\rho e^{i(\varphi-\theta)}\right|^{1+2\alpha}} . \tag{7} \]

From inequality (7) we have

\[ \int_{0}^{2\pi}\left|\lg |w_1(re^{i\varphi})|\right|\,d\mu(\varphi) \le c_6+c_9\int_{0}^{1}\frac{A_1(\rho)}{(1-\rho)^\alpha}\,d\rho \int_{0}^{2\pi} \frac{d\mu(\varphi)} {\left|1-re^{i(\varphi-\theta)}\right|^{1-\alpha}} \le c_{10}. \tag{8} \]

From inequalities (5) and (8), taking (2) into account, we obtain (4).
Applying Fatou’s lemma \((^4)\) in inequality (4), we obtain

\[ \int_{0}^{2\pi}\left|\lg |w(e^{i\varphi})|\right|\,d\mu(\varphi)<+\infty . \]

It follows from this that on a set \(E\) of positive \((1-\alpha)\)-capacity, \(f(e^{i\varphi})\) cannot be zero if \(f(z)\not\equiv 0\). The theorem is proved.

Theorem 1 is easily extended to meromorphic functions of the class \(T_\alpha\), if on the poles \(b_\nu\) of the function \(w(z)\) one imposes the condition

\[ \sum_{1}^{\infty}(1-|b_\nu|)^{1-\alpha}<+\infty . \]

\(4^\circ\). In \((^5)\) classes \(T_H\) of meromorphic functions, analogous to the classes \(T_\alpha\), are introduced as follows:

\(w(z)\in T_H\), if

\[ \int_{0}^{1} H(1-r)A(r)\,dr<+\infty \qquad \text{when } \int_{0}^{1}H(1-r)\,dr<\infty, \]

\[ \lim_{r\to 1-0} A(r)<+\infty \qquad \text{when } \int_{0}^{1}H(1-r)\,dr=\infty, \]

where \(H\) satisfies fairly general conditions.

For the classes \(T_H\), analogous results of L. Carleson are obtained \((^5)\) in terms of convex capacity \((^6)\).

We shall note only that if \(\lambda_n\) is the convex capacity of the set \(E\)—positive

\[ \left(\lambda_n=\sum_{k=n}^{\infty}\frac{1}{kH(1/k)}\right), \]

then, since, by Salem’s theorem \((^7)\), as \(x\to 0\)

\[ \sum_{1}^{\infty}\lambda_n\cos nx \sim \int_{1}^{1/x}\frac{du}{H(1/u)} \ge \frac{1}{xH(x)} \ge \frac{c_{11}}{h(x)}, \qquad \text{where } h(x)=\int_{0}^{x}H(u)\,du, \]

there exists a unit distribution \(\mu\) on \(E\) such that

\[ \int_{0}^{2\pi}\frac{d\mu(y)}{h(|x-y|)}<c_{12} \tag{9} \]

uniformly for \(0\le x\le 2\pi\).

If \(w(z)\in T_H\), then it can be represented in the form

\[ w(z)=B(z)w_1(z), \]

where \(w_1(z)\) belongs to the class \(T_H\) \((^5)\).

For the Blaschke product for which the series

\[ \sum_{1}^{\infty} h(1-|a_\nu|)<\infty \]

converges, a result analogous to (5) is obtained in the same way.

Let us further note that the representation of M. M. Dzhrbashyan remains valid for functions of the classes \(T_H\). We shall use this representation \({}^{(8)}\)

\[ w_1(z)=c_{13}\exp\left[\frac{1}{\pi}\int_0^1\int_0^{2\pi} \lg|w_1(z)|\,G(z\bar{\zeta})\,h(1-\rho)\,d\rho\,d\vartheta\right], \tag{10} \]

where \(z=re^{i\varphi}\), \(\zeta=\rho e^{i\vartheta}\), and the function \(G(z)\) has the form

\[ G(z)=\sum_0^\infty \frac{z^n}{\alpha^n}, \qquad \text{where } \alpha_n=O\left(\frac{1}{n}\,h\left(\frac{1}{n}\right)\right). \]

For the distribution \(\mu(\varphi)\) for which inequality (9) is satisfied, we obtain

\[ \int_0^{2\pi}|\lg|w_1(re^{i\varphi})||\,d\mu(\varphi)\leq \]

\[ \leq c_{14}+c_{15}\int_0^1 h(1-\rho)\,d\rho \int_0^{2\pi}|\lg|w_1(\rho e^{i\vartheta})||\,d\vartheta \int_0^{2\pi}\operatorname{Re}G(z\bar{\zeta})\,d\mu(\varphi). \]

Observing that from Salem’s theorem \({}^{(7)}\) one can obtain the inequality

\[ \sum_{n=0}^{\infty}\frac{n}{h(1/n)}\,\rho^n\cos nx \leq \frac{c_{16}}{h(x)(1-\rho)^2} \]

for the classes \(T_H\), analogously to Theorem 1 we can prove the following theorem:

Theorem 2. If an analytic function \(w(z)\in T_H\) in the disk and the zeros \(a_\nu\) of the function \(w(z)\) satisfy the condition

\[ \sum h(1-|a_\nu|)<\infty, \]

where

\[ h(t)=\int_0^t H(u)\,du, \]

then from the fact that the function \(w(z)\) has radial boundary values equal to zero on some set \(E\) whose \(\lambda_n\)-capacity is positive,

\[ \left(\lambda_n=\sum_{k=n}^{\infty}\frac{1}{kH(1/k)}\right), \]

it follows that \(w(z)\equiv0\) in \(|z|<1\).

For meromorphic functions \(w(z)\) of the class \(T_H\) the analogous result is true if one requires that the poles \(b_\nu\) of the function \(w(z)\) satisfy the condition

\[ \sum h(1-|b_\nu|)<+\infty. \]

I take this opportunity to express my sincere gratitude to Prof. M. M. Dzhrbashyan for his constant attention to the present work.

Institute of Mathematics and Mechanics
Academy of Sciences of the Armenian SSR

Received
14 XI 1963

REFERENCES

\({}^{1}\) I. I. Privalov, Boundary Properties of Analytic Functions, Moscow—Leningrad, 1950.
\({}^{2}\) M. M. Dzhrbashyan, Communications of the Institute of Mathematics and Mechanics, Academy of Sciences of the Armenian SSR, vol. 2, 3 (1958).
\({}^{3}\) L. Carleson, On a Class of Meromorphic Functions and its Associated Exceptional Sets, Uppsala, 1950.
\({}^{4}\) S. Saks, Theory of the Integral, Moscow, 1949.
\({}^{5}\) V. S. Zaharyan, Izv. Academy of Sciences of the USSR, Mathematical Series, 27, 801 (1963).
\({}^{6}\) K. V. Temko, Mat. sbornik, 51 (93), No. 2, 217 (1960).
\({}^{7}\) N. Bari, Trigonometric Series, Moscow, 1961.
\({}^{8}\) V. S. Zaharyan, Reports of the Academy of Sciences of the Armenian SSR, 36, No. 1, 3 (1963).