Abstract Generated abstract
This note extends Dzhrbashyan’s generalized Riemann Liouville type operator from a single weight function to a finite system of functions satisfying the class conditions for \(\Omega\). It defines associated coefficient sequences and kernels, then proves generalized Cauchy and Schwarz formulas for holomorphic functions after successive application of the operators. A corresponding generalized Poisson kernel is introduced, yielding an integral representation for harmonic functions in a disk in terms of the transformed boundary values on a smaller circle.
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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).