UDC 517.53

MATHEMATICS

I. I. BAVRIN

STRUCTURAL REPRESENTATION OF CERTAIN CLASSES OF HARMONIC AND ANALYTIC FUNCTIONS

(Presented by Academician M. A. Lavrent'ev on 19 I 1970)

M. M. Dzhrbashyan \((^1)\) gave a complete structural representation for the classes of harmonic and analytic functions associated with the operator \(L^{(\omega)}\) constructed by him \((^1)\). In the present note a complete structural representation is given for the classes of harmonic and analytic functions introduced here and associated with the operator \(L^{(\omega_1,\ldots,\omega_m)}\) \((^2)\).

Let \(U_{(\omega_1,\ldots,\omega_m)}\) denote the set of functions \(u(z)\) harmonic in the disk \(|z|<1\) and satisfying the condition

\[ \sup_{0<r<1}\left\{\int_0^{2\pi}\left|u_{(\omega_1,\ldots,\omega_m)}(re^{i\varphi})\right|\,d\varphi\right\}<+\infty, \]

where \(\omega_j(x)\in\Omega\) \((j=1,\ldots,m)\)* and

\[ u_{(\omega_1,\ldots,\omega_m)}(re^{i\varphi}) = L^{(\omega_1,\ldots,\omega_m)}[u(re^{i\varphi})]. \]

Let \(C_{(\omega_1,\ldots,\omega_m)}\) denote the class of functions \(f(z)\) analytic in the disk \(|z|<1\) and satisfying the condition

\[ \operatorname{Re} f_{(\omega_1,\ldots,\omega_m)}(z)\ge 0 \qquad (|z|<1), \]

where \(\omega_j(x)\in\Omega\) and
\[ f_{(\omega_1,\ldots,\omega_m)}(re^{i\varphi}) = L^{(\omega_1,\ldots,\omega_m)}[f(re^{i\varphi})]. \]

Let \(R_{(\omega_1,\ldots,\omega_m)}\) denote the class of functions \(f(z)\) analytic in the disk \(|z|<1\) for which the condition

\[ \sup_{0<r<1}\left\{\int_0^{2\pi} \left|\operatorname{Re} f_{(\omega_1,\ldots,\omega_m)}(re^{i\varphi})\right|\,d\varphi \right\}<+\infty. \]

is satisfied. Since, for \(f(z)\in C_{(\omega_1,\ldots,\omega_m)}\), the integrals

\[ \int_0^{2\pi} \left|\operatorname{Re} f_{(\omega_1,\ldots,\omega_m)}(re^{i\varphi})\right|\,d\varphi = \int_0^{2\pi} \operatorname{Re} f_{(\omega_1,\ldots,\omega_m)}(re^{i\varphi})\,d\varphi = 2\pi \operatorname{Re} f_{(\omega_1,\ldots,\omega_m)}(0) \]

are bounded by a number independent of \(r<1\), we obviously have

\[ C_{(\omega_1,\ldots,\omega_m)} \subset R_{(\omega_1,\ldots,\omega_m)}. \]

Let \(P(\theta,r;\omega_1,\ldots,\omega_m)\), \(S(z;\omega_1,\ldots,\omega_m)\) be the functions introduced by the author in \((^2)\).

* It is said (see \((^1)\), p. 1078) that a function \(\omega(x)\in\Omega\) if it is nonnegative and continuous on \([0,1)\), with
\[ \omega(0)=1,\qquad \int_0^1 \omega(x)\,dx<+\infty \]
and, for every \(r\) \((0\le r<1)\),
\[ \int_r^1 \omega(x)\,dx>0. \]

Theorem 1. a) The class \(U_{(\omega_1,\ldots,\omega_m)}\) coincides with the set of functions \(u(z)\) representable in the form of the integral

\[ u\left(re^{i\varphi}\right) = \frac{1}{2\pi} \int_{0}^{2\pi} P(\varphi-\theta,\ r;\ \omega_1,\ldots,\omega_m)\,d\psi(\theta) \tag{1} \]

\[ (0\le r<1,\ 0\le \varphi\le 2\pi), \]
where \(\psi(\theta)\) is an arbitrary real function with finite total variation on \([0,2\pi]\).

b) In representation (1) of a given function
\[ u(z)\in U_{(\omega_1,\ldots,\omega_m)} \]
the corresponding function \(\psi(\theta)\) can be determined by means of the limit

\[ \psi(\theta)= \lim_{n\to+\infty} \int_{0}^{\theta} u_{(\omega_1,\ldots,\omega_m)}\left(\rho_n e^{i\varphi}\right)\,d\varphi, \]

where \(\{\rho_n\}\), \(0<\rho_1<\rho_2<\cdots<\rho_n<\cdots\), \(\rho_n\uparrow 1\), is some increasing sequence.

c) The class
\[ U^{*}_{(\omega_1,\ldots,\omega_m)} \subset U_{(\omega_1,\ldots,\omega_m)} \]
of functions harmonic in the disk \(|z|<1\), for which

\[ u_{(\omega_1,\ldots,\omega_m)}(z)\ge 0 \qquad (|z|<1), \]

coincides with the set of functions representable in the form (1), where the function \(\psi(\theta)\) is nondecreasing on \([0,2\pi]\).

Theorem 2. a) The class \(C_{(\omega_1,\ldots,\omega_m)}\) coincides with the set of functions \(f(z)\) representable in the form

\[ f(z) = iC+ \frac{1}{2\pi} \int_{0}^{2\pi} S\left(e^{-i\theta}z;\ \omega_1,\ldots,\omega_m\right)\,d\psi(\theta) \tag{2} \]

\[ (|z|<1), \]
where \(\operatorname{Im} C=0\), and \(\psi(\theta)\) is an arbitrary nondecreasing bounded function on \([0,2\pi]\).

b) The class \(R_{(\omega_1,\ldots,\omega_m)}\) coincides with the set of functions representable in the form (2), where \(\psi(\theta)\) is a real function of finite variation on \([0,2\pi]\).

In the proof of Theorem 1, Theorem 2 of the author from \({}^{(2)}\) is used essentially, and in the proof of Theorem 2—the author’s Theorem 1 from \({}^{(2)}\) and Theorem 1 of the present note.

Moscow Regional Pedagogical Institute
named after N. K. Krupskaya

Received
6 I 1970

References

\({}^{1}\) M. M. Dzhrbashyan, Izv. AN SSSR, Ser. Mat., 32, No. 5, 1075 (1968).
\({}^{2}\) I. I. Bavrin, DAN, 187, No. 3 (1969).