Abstract
Full Text
Mathematics
N. A. Lebedev
The Area Principle in the Problem of Non-Overlapping Domains
(Presented by Academician V. I. Smirnov on 28 I 1960)
Let \(a_k,\ k=0,1,\ldots,n;\ n=0,1,\ldots,\) be given distinct points of the extended \(z\)-plane. Let \(D_k\) \((a_k\in D_k),\ k=0,1,\ldots,n,\) be arbitrary simply connected domains of the extended \(z\)-plane having no pairwise common points. Denote by \(w=f_k(z)\), \(f_k(0)=a_k,\ k=0,1,\ldots,n,\) a function mapping the disk \(|z|<1\) conformally and univalently onto the domain \(D_k\). In this way a system \(\{f_k(z)\}_0^n\) of \(n+1\) functions is obtained. The set of all such systems of functions will be called the class \(\mathfrak{M}(a_0,a_1,\ldots,a_n)\).
Let: \(R\) be the class of functions \(w=f(z)=\sum_{k=1}^{\infty} a_k z^k\), regular in the disk \(|z|<1\) and such that, for any points \(z_1\) and \(z_2\) in \(|z|<1\), the product \(f(z_1)f(z_2)\ne 1\); \(\Gamma\) be the class of functions \(w=f(z)=\sum_{k=1}^{\infty} a_k z^k\), regular in the disk \(|z|<1\) and such that, for any points \(z_1\) and \(z_2\) in \(|z|<1\), the product \(f(z_1)\overline{f(z_2)}\ne -1\); \(R^*\) and \(\Gamma^*\) are the subclasses of univalent functions respectively from the classes \(R\) and \(\Gamma\).
Let \(\{f_k(z)\}_0^n\in\mathfrak{M}(\infty,a_1,\ldots,a_n)\). Denote by \(D_k(r)\), \(k=0,1,\ldots,n\), the image of the disk \(|z|<r,\ 0<r<1,\) under the mapping by the function \(w=f_k(z)\), and by \(D(r)\) the complement in the extended \(w\)-plane of \(\bigcup_{k=0}^n \overline{D_k(r)}\).
\(1^\circ.\) Let \(\{f_k(z)\}_0^n\in\mathfrak{M}(\infty,a_1,\ldots,a_n)\). Let the function \(\xi=Q(w)\) be regular (and single-valued) in the domain \(D(r_0)\) for some \(r_0,\ 0<r_0<1\), and consequently the functions \(Q(f_l(z))\) are regular in the annulus \(r_0<|z|<1\) and are representable there in the form
\[ Q(f_l(z))=\sum_{q=1}^{\infty}\frac{\beta_q^{(l)}}{z^q}+\sum_{q=0}^{\infty} b_q^{(l)}z^q . \]
Lemma. The inequality holds
\[ \sum_{l=0}^{n}\sum_{q=1}^{\infty} q\left|b_q^{(l)}\right|^2 \le \sum_{l=0}^{n}\sum_{q=1}^{\infty} q\left|\beta_q^{(l)}\right|^2 = A. \tag{1} \]
To prove this inequality it is necessary to compute the area \(S(r)\) of the image of the domain \(D(r)\), \(r_0<r<1\), under the mapping by the function \(\xi=Q(w)\) (see \((^{1,2})\)) and let \(r\) tend to one. Equality in (1) holds if and only if \(\lim_{r\to 1} S(r)=0\).
If, in the statement of the lemma, one requires only that the function \(Q(w)\) have a regular and (single-valued) derivative in \(D(r_0)\), then in the right-hand side of inequality (1) there will appear some additional term, owing to the fact that in the expansion of the function \(Q(f_l(z))\) in the annulus \(r_0<|z|<1\) there will appear a term \(\beta^{(l)}\ln z\).
Corollary. Let \(C_q^{(l)}\) be arbitrary numbers such that the series
\(\sum_{q=1}^{\infty} q |C_q^{(l)}|\), \(l=0,1,\ldots,n\), converge. If the conditions of the lemma are satisfied, the following inequalities hold:
\[ \sum_{l=0}^{n}\lambda_l\left|\sum_{q=1}^{\infty}q b_q^{(l)}C_q^{(l)}\right| \leq A-\sum_{l=0}^{n}\sum_{q=1}^{\infty}q\left|b_q^{(l)}-\lambda_l e^{i\theta_l}C_q^{(l)}\right|^2, \tag{2} \]
where
\[ \lambda_l= \left|\sum_{q=1}^{\infty}q b_q^{(l)}C_q^{(l)}\right| \cdot \left(\sum_{q=1}^{\infty}q|C_q^{(l)}|^2\right)^{-1}, \qquad \theta_l=\arg\sum_{q=1}^{\infty}q b_q^{(l)}C_q^{(l)}; \]
\[ \left(\sum_{l=0}^{n}\left|\sum_{q=1}^{\infty}q b_q^{(l)}C_q^{(l)}\right|\right)^2 \leq \left(A-\sum_{l=0}^{n}\sum_{q=1}^{\infty}q\left|b_q^{(l)}-\lambda e^{i\theta_l}C_q^{(l)}\right|^2\right) \sum_{l=0}^{n}\sum_{q=1}^{\infty}q|C_q^{(l)}|^2, \tag{3} \]
where
\[ \lambda= \left(\sum_{l=0}^{n}\left|\sum_{q=1}^{\infty}q b_q^{(l)}C_q^{(l)}\right|\right) \cdot \left(\sum_{l=0}^{n}\sum_{q=1}^{\infty}q|C_q^{(l)}|^2\right)^{-1}, \qquad \theta_l=\arg\sum_{q=1}^{\infty}q b_q^{(l)}C_q^{(l)}. \]
Equality in inequalities (2) and (3) holds if and only if \(\lim_{r\to 1} S(r)=0\).
\(2^\circ.\) Let \(z_{\nu,k}, z'_{\nu,k}\) \((\nu=0,1,\ldots,m;\ k=0,1,\ldots,n)\) be arbitrary points from the disk \(|z|<1\); let \(\gamma_{\nu,k}, \gamma'_{\nu,k}\) \((\nu=0,1,\ldots,m;\ k=0,1,\ldots,n)\), \(x_{\nu,k}, x'_{\nu,k}\) \((\nu=0,1,\ldots,m,\ k=0,1,\ldots,n)\) be arbitrary numbers.
Let \(\{f_k(z)\}_0^n\in \mathfrak M(\infty,a_1,\ldots,a_n)\). Consider functions \(\psi_k(w,\xi)\), \(k=0,1,\ldots,n\), such that the function \(\psi_k(w,\xi)\), for every fixed \(\xi\in D_k(r)\), \(0<r<1\), is regular in \(w\) outside \(\overline{D}_k(r)\) and, for every fixed \(w\in \overline{D}_k(r)\), is regular in \(D_k(r)\) with respect to \(\xi\). It is clear that for \(k\ne l\)
\[ \psi_k(f_l(z),f_k(\xi)) = \sum_{q=0}^{\infty}\sum_{p=0}^{\infty} b_{p,q}^{k,l}\xi^p z^q = \sum_{q=0}^{\infty} b_q^{k,l}(\xi) z^q = b_0^{k,l}(\xi)+\varphi_{k,l}(\xi,z) \]
for \(|z|<1,\ |\xi|<1\);
\[ \psi_l(f_l(z),f_l(\xi)) = \sum_{q=0}^{\infty}\sum_{p=0}^{\infty} b_{p,q}^{l,l}\xi^p z^q + \sum_{q=1}^{\infty}\sum_{p=0}^{\infty} \beta_{p,q}^{l,l}\xi^p z^{-q} = \]
\[ = \sum_{q=0}^{\infty} b_q^{l,l}(\xi) z^q + \sum_{q=1}^{\infty} \beta_q^{l,l}(\xi) z^{-q} = b_0^{l,l}(\xi)+\varphi_{l,l}(\xi,z)+\sum_{q=1}^{\infty}\beta_q^{l,l}(\xi)z^{-q} \]
in the domain \(|\xi|<|z|<1\).
Introduce the function
\[ Q(w)=\sum_{k=0}^{n}\sum_{\nu=0}^{m}\gamma_{\nu,k}\psi_k(w,f_k(z_{\nu,k})). \]
This function satisfies the conditions of the lemma. Consequently, the following inequalities hold:
\[ \sum_{l=0}^{n}\sum_{q=1}^{\infty} q \left|\sum_{k=0}^{n}\sum_{\nu=0}^{m}\gamma_{\nu,k} b_q^{k,l}(z_{\nu,k})\right|^2 \leq \sum_{l=0}^{n}\sum_{q=1}^{\infty} q \left|\sum_{\nu=0}^{m}\gamma_{\nu,l}\beta_q^{l,l}(z_{\nu,l})\right|^2 = A'; \tag{1'} \]
\[ \sum_{l=0}^{n}\frac{1}{A_l'} \left| \sum_{k=0}^{n}\sum_{\nu,\nu'=0}^{m} \gamma_{\nu,k}\gamma'_{\nu',l} \frac{\partial^s}{\partial z_{\nu,k}^{\,s}} \varphi_{k,l}(z_{\nu,k},z'_{\nu',l}) \right| \leq A', \qquad s=0,1,\ldots; \tag{2'} \]
\[ \sum_{l=0}^{n}\left|\sum_{k=0}^{n}\sum_{\nu,\nu'=0}^{m} \gamma_{\nu,k}\gamma'_{\nu',l}\, \frac{\partial^s}{\partial z_{\nu,k}^s}\, \varphi_{k,l}\left(z_{\nu,k},z'_{\nu',l}\right)\right|^2 \leq \left[A'\sum_{l=0}^{n}A'_l\right]^{1/2}, \qquad s=0,1,\ldots, \tag{3'} \]
where
\[ A'_l=\sum_{\nu,\nu'=0}^{m}\gamma'_{\nu,l}\overline{\gamma'_{\nu',l}} \left[ \frac{\partial^{2s}}{\partial z^s\partial \zeta^s} \ln\frac{1}{1-z\zeta} \right]_{\zeta=z'_{\nu,l};\,z=\overline{z'_{\nu',l}}}. \]
To obtain inequalities \((2')\) and \((3')\), one must apply the corollary of the lemma to inequality \((1')\), putting
\[ C_q^{(l)}=\frac{1}{q}\sum_{\nu=0}^{m}\gamma'_{\nu,l}\, \frac{q!}{(q-s)!}\,(z'_{\nu,l})^{q-s} \]
for \(q\geq s\), and \(C_q^{(l)}=0\) for \(q<s\).
Putting in inequalities \((1')\), \((2')\), and \((3')\)
\[ \gamma_{\nu,k}=\frac{1}{\overline m+1}\sum_{p=0}^{\overline m}x_{p,k}\overline z_{\nu,k}^{\,p}, \]
\[ \gamma'_{\nu,l}=\frac{1}{\overline m+1}\sum_{q=0}^{\overline m}x'_{q,l}(\overline z_{\nu,l})^q, \qquad z_{\nu,l}=z'_{\nu,l}=re^{i\theta_\nu}, \qquad \theta_\nu=\frac{2\pi}{m+1}\nu,\quad 0<r<1, \]
and letting \(m\) tend to \(\infty\), and then \(r\) to unity, in the limit we obtain the inequalities
\[ \sum_{l=0}^{n}\sum_{q=1}^{\infty}q \left|\sum_{k=0}^{n}\sum_{p=0}^{\overline m} b_{p,q}^{k,l}x_{p,k}\right|^2 \leq \sum_{l=0}^{p}\sum_{q=1}^{\infty}q \left|\sum_{p=0}^{\overline m}\beta_{p,q}^{l,l}x_{p,l}\right|^2 =A'', \tag{1''} \]
\[ \sum_{l=0}^{n}\frac{1}{A''_l} \left|\sum_{k=0}^{n}\sum_{p,q=0}^{\overline m} b_{p,q}^{k,l}\frac{(q+s)!}{q!}\,x_{p,k}x'_{q,l}\right|^2 \leq A'', \qquad s=0,1,\ldots; \tag{2''} \]
\[ \sum_{l=0}^{n} \left|\sum_{k=0}^{n}\sum_{p,q=0}^{\overline m} b_{p,q+s}^{k,l}\frac{(q+s)!}{q!}\,x_{q,k}x'_{q,l}\right| \leq \left[A''\sum_{l=0}^{n}A''_0\right]^{1/2}, \qquad s=0,1,\ldots, \tag{3''} \]
where
\[ A''_l=\sum_{q=0}^{\overline m}\frac{(q+s)!}{q!}\,\frac{1}{q}\,|x'_{q,l}|^2. \]
In inequalities \((2'')\) and \((3'')\), for \(s=0\) the summation over \(q\) must run from \(1\) to \(\overline m\) (and not from zero to \(\overline m\)).
Putting in inequalities \((1'')\), \((2'')\), and \((3'')\)
\[ x_{p,k}=\sum_{\nu=0}^{m}\gamma_{\nu,k}z_{\nu,k}^{p}, \qquad x'_{q,l}=\sum_{\nu=0}^{m}\gamma'_{\nu,l}(z'_{\nu,l})^q \]
and letting \(\overline m\) tend to \(\infty\), in the limit we obtain inequalities \((1')\), \((2')\), and \((3')\).
\(3^\circ\). As the functions \(\psi_k(w,\xi)\) occurring in item \(2^\circ\), one may take, for example,
\[ \psi_k(w,\xi)= \begin{cases} \displaystyle \ln\left(1-\frac{\xi-a_k}{w-a_k}\right), & k=1,2,\ldots,n,\\[1.2ex] \displaystyle \ln\left(1-\frac{w}{\xi}\right), & k=0. \end{cases} \]
In this case
\[ A'=\sum_{l=0}^{n}\sum_{\nu,\nu'=0}^{m} \gamma_{\nu,l}\overline{\gamma_{\nu',l}}\, \ln\frac{1}{1-z_{\nu,l}\overline z_{\nu',l}} = \sum_{l=0}^{n}\sum_{\nu,\nu'=0}^{m} \gamma_{\nu,l}\overline{\gamma_{\nu',l}} \sum_{q=1}^{\infty}\frac{1}{q}(z_{\nu,l}\overline z_{\nu',l})^q, \]
\[ A''=\sum_{l=0}^{n}\sum_{q=1}^{m}\frac{1}{q}|x_{q,l}|^2. \]
\(4^\circ\). In the case considered in item \(3^\circ\), the inequalities \((1')\), \((2')\), \((3')\), \((1'')\), \((2'')\), \((3'')\) can be generalized. In this case, for example, inequality \((2)\) (for \(s=0\)) will correspond to the inequality
\[ \sum_{l=0}^{n}\frac{1}{A_l^{(s)}} \left| \sum_{k=0}^{n}\sum_{\nu,\nu'=0}^{m} \gamma_{\nu,k}'\bar{\gamma}_{\nu',l}\, \varphi_{k,l}^{s}(z_{\nu,k}',z_{\nu',l}) \right|^{2} \leq \sum_{l=0}^{n}\sum_{\nu,\nu'=0}^{m} \gamma_{\nu,l}'\bar{\gamma}_{\nu',l} \ln^{s}\frac{1}{1-z_{\nu,l}'\bar z_{\nu',l}}, \qquad s=1,2,\ldots, \]
where
\[ A_l^{(s)}= \sum_{\nu,\nu'=0}^{m} \gamma_{\nu,l}'\bar{\gamma}_{\nu',l} \ln^{s}\frac{1}{1-z_{\nu,l}'\bar z_{\nu',l}}. \]
Further, it is easy to obtain six more inequalities. In this case, to the preceding inequality there will correspond the inequality
\[ \sum_{l=0}^{n}\frac{1}{A_l^{*}} \left| \sum_{k=0}^{m}\sum_{\nu,\nu'=0}^{m} \gamma_{\nu,k}'\bar{\gamma}_{\nu',l}\, e^{\varphi_{k,l}(z_{\nu,k},z_{\nu',l}')} \right|^{2} \leq \sum_{l=0}^{n}\sum_{\nu,\nu'=0}^{m} \frac{\gamma_{\nu,l}'\bar{\gamma}_{\nu',l}}{1-z_{\nu,l}'\bar z_{\nu',l}'}, \]
\[ A_l^{*}= \sum_{\nu,\nu'=0}^{m} \frac{\gamma_{\nu,l}'\bar{\gamma}_{\nu',l}}{1-z_{\nu,l}'\bar z_{\nu',l}'}. \]
The latter inequalities make it possible to obtain a number of interesting integral inequalities. In particular, the following holds.
Theorem. If \(\{f_0(z), f_1(z)\}\in \mathfrak M(\infty,0)\), then the inequality
\[ \frac{1}{2\pi}\int_{0}^{2\pi}|f_1(e^{i\theta})|^{2}\,d\theta\cdot \frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{|f_0(e^{i\theta})|^{2}}\,d\theta \leq 1. \]
holds.
The equality sign holds if and only if
\[ f_0(z)=\frac{a}{z}+b,\qquad |a|>|b|,\qquad f_1(z)=\frac{(|a|^{2}-|b|^{2})\eta z}{\bar a-\bar b\eta z}, \qquad |\eta|=1. \]
Corollary. If \(f(z)=\sum_{k=1}^{\infty}a_k z^k\in R^{*}\) (or \(\Gamma^{*}\)), then the inequality
\[ \frac{1}{2\pi}\int_{0}^{2\pi}|f(e^{i\theta})|^{2}\,d\theta = \sum_{k=1}^{\infty}|\alpha_k|^{2} \leq 1. \tag{4} \]
holds.
The equality sign holds if and only if
\[ f(z)=\frac{\eta z}{R\pm \sqrt{R^{2}-1}\,\eta z}, \qquad R\geq 1,\qquad |\eta|=1. \]
Inequality (4) also holds in the case when \(f(z)\in R\) (or \(\Gamma\)). It strengthens the result obtained in paper \((1)\):
\[ \frac{1}{2\pi}\int_{0}^{2\pi}|f(e^{i\theta})|\,d\theta\leq 1, \qquad f(z)\in R\ \text{(or }\Gamma\text{)}. \]
Received
26 I 1960
CITED LITERATURE
- N. A. Lebedev, I. M. Milin, Matem. sbornik, 28(70), No. 2, 359 (1951).
- N. A. Lebedev, Some estimates and extremum problems in conformal mapping, Candidate dissertation, LSU, 1951.