Full Text
UDC 517.53
MATHEMATICS
I. I. Bavrin
ON A GENERALIZATION OF THE INTEGRAL FORMULAS OF CAUCHY, SCHWARZ, AND POISSON
(Presented by Academician M. A. Lavrent′ev on 5 February 1969)
M. M. Dzhrbashyan \((^{1})\) constructed a generalized operator \(L^{(\omega)}\) of Riemann–Liouville type, by means of which he established fundamentally new analogues of the classical formulas of Cauchy, Schwarz, and Poisson*—generalized Cauchy, Schwarz, and Poisson formulas associated with a given function \(\omega(x)\in\Omega\)**. In the present note this generalized operator is used to establish (Theorems 1, 2) generalized Cauchy, Schwarz, and Poisson formulas associated with a given system of functions \(\omega_j(x)\in\Omega\) \((j=1,2,\ldots,m)\).
Let the functions \(\omega_j(x)\in\Omega\) \((j=1,2,\ldots,m)\). Further, let \(p_j(0)=1\),
\[ p_j(r)=r\int_r^1 \frac{\omega_j(x)}{x^2}\,dx \quad (r\in(0,1]), \qquad \Delta_0^{(j)}=1,\quad \Delta_k^{(j)}=-(k+1)\int_0^1 r^k\,dp_j(r)= \]
\[ =k\int_0^1 r^{k-1}\omega_j(r)\,dr \quad (j=1,2,\ldots,m),\quad k=1,2,\ldots^{***}. \]
We introduce for consideration the power series
\[ C(z;\omega_1,\ldots,\omega_m)= \sum_{k=0}^{\infty}\frac{z^k}{\Delta_k^{(1)}\cdots\Delta_k^{(m)}}. \tag{1} \]
It is easy to see that the radius of convergence of this series is equal to one. Thus the function \(C(z;\omega_1,\ldots,\omega_m)\) is holomorphic in the disk \(|z|<1\). Along with this function we also introduce the function
\[ S(z;\omega_1,\ldots,\omega_m) =2C(z;\omega_1,\ldots,\omega_m)-C(0;\omega_1,\ldots,\omega_m)= \]
\[ =1+2\sum_{k=1}^{\infty}\frac{z^k}{\Delta_k^{(1)}\cdots\Delta_k^{(m)}}, \tag{2} \]
noting that \(C(0;\omega_1,\ldots,\omega_m)=1/\Delta_0^{(1)}\cdots\Delta_0^{(m)}=1\).
* For other generalizations of the Cauchy, Schwarz, and Poisson formulas, see, for example, \((^{2-6})\).
** It is said (see \((^{1})\), p. 1078) that a function \(\omega(x)\in\Omega\) if it is nonnegative and continuous on \([0,1)\), with \(\omega(0)=1\),
\[ \int_0^1 \omega(x)\,dx<+\infty \]
and for every \(r\) \((0\leq r<1)\)
\[ \int_r^1 \omega(x)\,dx>0. \]
*** In \((^{1})\) the function \(p(0)=1\),
\[ p(r)=r\int_r^1 \frac{\omega(x)}{x^2}\,dx \quad (\omega(x)\in\Omega),\quad r\in(0,1], \]
and the sequence of numbers
\[ \Delta_k=-(k+1)\int_0^1 r^k\,dp(r)\quad (k=0,1,2,\ldots) \]
were introduced, and it was shown that all the numbers \(\Delta_k\) \((k=0,1,2,\ldots)\) are positive, with \(\Delta_0=1\),
\[ \Delta_k=k\int_0^1 \omega(x)x^{k-1}\,dx\quad (k=1,2,\ldots). \]
Theorem 1. Let the function
\[ f\left(re^{i\varphi}\right)=\sum_{k=0}^{\infty} a_k\left(re^{i\varphi}\right)^k \]
be holomorphic in the disk \(|z|<R\). Then the function
\[ L^{(\omega_m)}\left[L^{(\omega_{m-1})}\ldots\left[L^{(\omega_1)}\left[f\left(re^{i\varphi}\right)\right]\right]\ldots\right]\equiv \]
\[ \equiv L^{(\omega_1,\ldots,\omega_m)}\left[f\left(re^{i\varphi}\right)\right] \equiv f_{(\omega_1,\ldots,\omega_m)}\left(re^{i\varphi}\right) =\sum_{k=0}^{\infty}\Delta_k^{(1)}\ldots\Delta_k^{(m)}a_k\left(re^{i\varphi}\right)^k \tag{3} \]
is holomorphic in the same disk \(|z|<R\), and for any \(\rho\) \((0<\rho<R)\) the integral formulas
\[ f(z)=\frac{1}{2\pi}\int_{0}^{2\pi} C\left(e^{-i\theta}\frac{z}{\rho};\omega_1,\ldots,\omega_m\right) f_{(\omega_1,\ldots,\omega_m)}\left(\rho e^{i\theta}\right)d\theta \quad (|z|<\rho), \]
\[ f(z)=i\operatorname{Im} f(0)+\frac{1}{2\pi}\int_{0}^{2\pi} S\left(e^{-i\theta}\frac{z}{\rho};\omega_1,\ldots,\omega_m\right)\times \]
\[ \times \operatorname{Re} f_{(\omega_1,\ldots,\omega_m)}\left(\rho e^{i\theta}\right)d\theta \quad (|z|<\rho). \]
In the course of the proof, the Cauchy–Hadamard formula is essentially used, as well as the formulas
\[ L^{(\omega_1,\ldots,\omega_m)}[r^k]=\Delta_k^{(1)}\ldots\Delta_k^{(m)}r^k \quad (k=0,1,2,\ldots) \]
and the expansions (1), (2), and (3).
Let us introduce into consideration the function
\[ P(\theta,r;\omega_1,\ldots,\omega_m) =\operatorname{Re}S\left(re^{i\theta};\omega_1,\ldots,\omega_m\right) =1+2\sum_{k=1}^{\infty} \frac{r^k\cos k\theta}{\Delta_k^{(1)}\ldots\Delta_k^{(m)}}, \]
harmonic in the unit disk \(0\le r<1,\ 0\le\theta\le 2\pi\).
From Theorem 1 it follows easily:
Theorem 2. Let \(u(z)\) be a harmonic function in the disk \(|z|<R\). Then the function
\[ u_{(\omega_1,\ldots,\omega_m)}\left(re^{i\varphi}\right) = L^{(\omega_1,\ldots,\omega_m)} \left[u\left(re^{i\varphi}\right)\right] \]
will be harmonic in the same disk \(|z|<R\), and for any \(\rho\) \((0<\rho<R)\) the integral formula
\[ u\left(re^{i\varphi}\right) = \frac{1}{2\pi}\int_{0}^{2\pi} P\left(\varphi-\theta,\frac{r}{\rho};\omega_1,\ldots,\omega_m\right) u_{(\omega_1,\ldots,\omega_m)}\left(\rho e^{i\theta}\right)d\theta \]
holds,
\[ (0\le r<\rho,\qquad 0\le\varphi\le 2\pi). \]
Moscow Regional Pedagogical Institute
named after N. K. Krupskaya
Received
11 XII 1968
REFERENCES
- M. M. Dzhrbashyan, Izv. Akad. Nauk SSSR, Ser. Mat., 32, No. 5, 1075 (1968).
- M. M. Dzhrbashyan, Integral transformations and representations of functions in the complex domain, Moscow, 1966.
- I. I. Bavrin, DAN, 172, No. 6 (1967).
- I. I. Bavrin, DAN, 176, No. 6 (1967).
- I. I. Bavrin, DAN, 180, No. 1 (1968).
- I. I. Bavrin, DAN, 186, No. 2 (1969).