Full Text
UDC 517.53
MATHEMATICS
I. I. BAVRIN
ON THE QUESTION OF GENERALIZING THE INTEGRAL FORMULAS OF CAUCHY, SCHWARZ, AND POISSON
(Presented by Academician M. A. Lavrent′ev on March 6, 1970)
M. M. Dzhrbashyan \((^{1})\) established generalized formulas of Cauchy, Schwarz, and Poisson associated with a given function \(\omega(x)\in\Omega\). Subsequently the author \((^{2})\) obtained generalized formulas of Cauchy, Schwarz, and Poisson associated with a given system of functions \(\omega_j(x)\in\Omega\) \((j=1,\ldots,m)\). In the present note a substantial generalization of these latter formulas is given (Theorems 1 and 2).
- Let the functions \(\omega_j(x)\in\Omega\) \((j=1,2,\ldots,m)\), \(\widetilde{\omega}_{\tilde j}(x)\in\Omega\) \((\tilde j=1,2,\ldots,\tilde m)\). Suppose, further, that
\[ p_j(0)=1,\qquad \widetilde{p}_{\tilde j}(0)=1, \]
\[ p_j(r)=r\int_r^1\frac{\omega_j(x)}{x^2}\,dx,\qquad \widetilde{p}_{\tilde j}(r)=r\int_r^1\frac{\widetilde{\omega}_{\tilde j}(x)}{x^2}\,dx \quad (r\in(0,1)), \]
\[ \Delta_0^{(j)}=1,\qquad \widetilde{\Delta}_0^{(\tilde j)}=1,\qquad \Delta_k^{(j)}=-(k+1)\int_0^1 r^k\,dp_j(r) = k\int_0^1 r^{k-1}\omega_j(r)\,dr, \]
\[ \widetilde{\Delta}_k^{(\tilde j)} =-(k+1)\int_0^1 r^k\,d\widetilde{p}_{\tilde j}(r) = k\int_0^1 r^{k-1}\widetilde{\omega}_{\tilde j}(r)\,dr \]
\[ (j=1,2,\ldots,m;\quad \tilde j=1,2,\ldots,\tilde m),\qquad k=1,2,\ldots\ .* \]
- In \((^{2})\) the author introduced into consideration the function
\[ C(z;\omega_1,\ldots,\omega_m) =\sum_{k=0}^{\infty}\frac{z^k}{\Delta_k^{(1)}\cdots \Delta_k^{(m)}}, \]
which is holomorphic in the disk \(|z|<1\). Therefore, by virtue of Theorem 1 of \((^{2})\), the function
\[ L_{(\widetilde{\omega}_1,\ldots,\widetilde{\omega}_{\tilde m})} \left[C(re^{i\varphi};\omega_1,\ldots,\omega_m)\right] \equiv C_{(\widetilde{\omega}_1,\ldots,\widetilde{\omega}_{\tilde m})} (re^{i\varphi};\omega_1,\ldots,\omega_m) = \]
\[ =\sum_{k=0}^{\infty} \frac{\widetilde{\Delta}_k^{(1)}\cdots \widetilde{\Delta}_k^{(\tilde m)}} {\Delta_k^{(1)}\cdots \Delta_k^{(m)}}(re^{i\varphi})^k, \tag{1} \]
* In \((^{1})\) the function
\[ p(r)=r\int_r^1\frac{\omega(x)}{x^2}\,dx \quad (\omega(x)\in\Omega),\quad r\in(0,1),\quad p(0)=1, \]
and the sequence of numbers
\[ \Delta_k=-(k+1)\int_0^1 r^k\,dp(r)\quad (k=0,1,2,\ldots) \]
were introduced; moreover, it was shown that all the numbers \(\Delta_k\) \((k=0,1,2,\ldots)\) are positive, with
\[ \Delta_0=1,\qquad \Delta_k=k\int_0^1 \omega(x)x^{k-1}\,dx \quad (k=1,2,\ldots). \]
which, for brevity, we shall denote by \(C_{(\widetilde{\omega})}(re^{i\varphi};\omega)\) \((\omega=(\omega_1,\ldots,\omega_m),\ \widetilde{\omega}=(\widetilde{\omega}_1,\ldots,\widetilde{\omega}_{\widetilde m})\), and similarly everywhere below), is holomorphic in the disk \(|z|<1\).
Introduce the function
\[ S_{(\widetilde{\omega})}(z;\omega)=2C_{(\widetilde{\omega})}(z;\omega)-C_{(\widetilde{\omega})}(0;\omega) =1+2\sum_{k=1}^{\infty} \frac{\widetilde{\Delta}^{(1)}_k\cdots \widetilde{\Delta}^{(\widetilde m)}_k} {\Delta^{(1)}_k\cdots \Delta^{(m)}_k}z^k, \tag{2} \]
noting that
\[ C_{(\widetilde{\omega})}(0;\omega):= \widetilde{\Delta}^{(1)}_0\cdots \widetilde{\Delta}^{(\widetilde m)}_0 / \Delta^{(1)}_0\cdots \Delta^{(m)}_0 =1. \]
Let the function
\[ f(re^{i\varphi})=\sum_{k=0}^{\infty} a_k(re^{i\varphi})^k \]
be holomorphic in the disk \(|z|<R\). Then, by ((2), Theorem 1), the function
\[ L^{(\omega_1,\ldots,\omega_m)}[f(re^{i\varphi})]\equiv f_{(\omega_1,\ldots,\omega_m)}(re^{i\varphi}) = \sum_{k=0}^{\infty} \Delta^{(1)}_k\cdots \Delta^{(m)}_k a_k(re^{i\varphi})^k \]
is holomorphic in the same disk \(|z|<R\). Hence it follows that the function
\[ f_{(\omega)}(z;\widetilde{\omega}):= f_{(\omega_1,\ldots,\omega_m)}(z;\widetilde{\omega}_1,\ldots,\widetilde{\omega}_{\widetilde m}) = \sum_{k=0}^{\infty} \frac{\Delta^{(1)}_k\cdots \Delta^{(m)}_k} {\widetilde{\Delta}^{(1)}_k\cdots \widetilde{\Delta}^{(\widetilde m)}_k} a_k z^k \tag{3} \]
is also holomorphic in the disk \(|z|<R\)*. Finally, relying on the expansions (1)—(3), we arrive at the following theorem.
Theorem 1. If the function
\[ f(z)=\sum_{k=0}^{\infty}a_k z^k \]
is holomorphic in the disk \(|z|<R\), then for any \(\rho\) \((0<\rho<R)\) the integral formulas
\[ f(z)=\frac{1}{2\pi}\int_{0}^{2\pi} C_{(\widetilde{\omega})}\left(e^{-i\theta}\frac{z}{\rho};\omega\right) f_{(\omega)}(\rho e^{i\theta};\widetilde{\omega})\,d\theta \qquad (|z|<\rho), \]
\[ f(z)= i\,\operatorname{Im} f(0)+\frac{1}{2\pi}\int_{0}^{2\pi} S_{(\widetilde{\omega})}\left(e^{-i\theta}\frac{z}{\rho};\omega\right) \operatorname{Re} f_{(\omega)}(\rho e^{i\theta};\widetilde{\omega})\,d\theta \qquad (|z|<\rho). \]
- Introduce the function
\[ P_{(\widetilde{\omega})}(\theta,r;\omega) =\operatorname{Re} S_{(\widetilde{\omega})}(re^{i\theta};\omega) = 1+2\sum_{k=1}^{\infty} \frac{\widetilde{\Delta}^{(1)}_k\cdots \widetilde{\Delta}^{(\widetilde m)}_k} {\Delta^{(1)}_k\cdots \Delta^{(m)}_k} r^k\cos k\theta, \]
which is harmonic in the unit disk \(0\le r<1,\ 0\le \theta\le 2\pi\).
Let the function
\[ u(re^{i\varphi})=\alpha_0+ \sum_{k=1}^{\infty}(\alpha_k\cos k\varphi-\beta_k\sin k\varphi)r^k \]
\((\alpha_0,\alpha_k,\beta_k\ (k=1,2,\ldots)\) are real numbers**\()\) be harmonic in the disk \(|z|<R\). Then, by virtue of the author’s theorem (Theorem 2 from (?)), the function
\[ L^{(\omega_1,\ldots,\omega_m)}[u(re^{i\varphi})]\equiv u_{(\omega_1,\ldots,\omega_m)}(re^{i\varphi}) = \alpha_0+ \sum_{k=1}^{\infty} (\Delta^{(1)}_k\cdots \Delta^{(m)}_k\alpha_k\cos k\varphi- \]
\[ -\Delta^{(1)}_k\cdots \Delta^{(m)}_k\beta_k\sin k\varphi)r^k \]
* The Cauchy–Hadamard formula is used.
** They remain so in what follows as well.
will be harmonic in the same disk \(|z|<R\). Hence it follows that the function
\[ u_{(\omega)}(re^{i\varphi};\widetilde{\omega}) = u_{(\omega_1,\ldots,\omega_m)}(re^{i\varphi};\widetilde{\omega}_1,\ldots,\widetilde{\omega}_m) = \]
\[ = \alpha_0 + \sum_{k=1}^{\infty} \left( \frac{\Delta_k^{(1)}\cdots \Delta_k^{(m)}}{\widetilde{\Delta}_k^{(1)}\cdots \widetilde{\Delta}_k^{(m)}}\alpha_k\cos k\varphi - \frac{\Delta_k^{(1)}\cdots \Delta_k^{(m)}}{\widetilde{\Delta}_k^{(1)}\cdots \widetilde{\Delta}_k^{(m)}}\beta_k\sin k\varphi \right)r^k \]
is also harmonic in the disk \(|z|<R\).* Now Theorem 1 easily implies
Theorem 2. If the function
\[ u(re^{i\varphi}) = \alpha_0 + \sum_{k=1}^{\infty} (\alpha_k\cos k\varphi-\beta_k\sin k\varphi)r^k \]
is harmonic in the disk \(|z|<R\), then for every \(\rho\) \((0<\rho<R)\) the integral formula
\[ u(re^{i\varphi}) = \frac{1}{2\pi} \int_0^{2\pi} P_{(\widetilde{\omega})} \left(\varphi-\theta,\frac{r}{\rho};\omega\right) u_{(\omega)}(\rho e^{i\theta};\widetilde{\omega})\,d\theta \]
\[ (0\le r<\rho,\qquad 0\le \varphi\le 2\pi). \]
is valid.
Moscow Regional Pedagogical Institute
named after N. K. Krupskaya
Received
5 III 1970
CITED LITERATURE
¹ M. M. Dzhrbashyan, Izv. AN SSSR, ser. matem., 32, No. 5, 1075 (1968). ² I. I. Bavrin, DAN, 187, No. 3 (1969).
* The Cauchy–Hadamard formula is used.