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MATHEMATICS
On small variation formulas
Ya. V. Borisova,
I. A. Kolesnikov,
S. A. Kopanev Tomsk State University, Tomsk, Russian Federation
Abstract:
One of the main methods for solving extremal problems is the variational method. Variational
formulas are the main tool of the variational method. Some variational formulas, the so-called
small variational formulas, were obtained by means of a family of mappings from the unit disk
onto domains lying in the unit disk. There is a theorem in the paper that gives a rather general
approach to obtaining small variational formulas.
Theorem. Let the map
$g: E_z\times (0,\varepsilon_0)\to E_\zeta$,
$\zeta=g(z,\varepsilon)$ satisfy the following conditions:
- $\forall \varepsilon\in (0,\varepsilon_0)$, the contraction $g\mid_{E\times\{\varepsilon\}}$ is a holomorphic univalent mapping;
- $\lim\limits_{\varepsilon\to+0}g(z,\varepsilon)=z$, locally uniformly in $E_z$;
- there exists a partial right derivative of $g(z,\varepsilon)$ and $g'_z(z,\varepsilon)$ with respect to $\varepsilon$ at the origin,
locally uniformly in $E_z$.
Then, in the class
$S$ for the mapping
$f\in S$, the following variational formulas take place:
\begin{gather*}
f_1(z,\varepsilon)=f(z)+\varepsilon\left(f'(z)g'_\varepsilon(z,0) -f(z)f''(0)g'_\varepsilon(0,0)-f(z)g''_{z\varepsilon}(0,0)\right)+o(z,\varepsilon),\\ \varepsilon\in(0,\varepsilon_0), \tag{1}
\end{gather*}
where $\lim\limits_{\varepsilon\to+0}\frac{o(z,\varepsilon)}{\varepsilon}=0$ locally uniformly in
$E_z$;
\begin{gather*}
f_2(z,\varepsilon)=f(z)+\varepsilon\left(f'(z)\left(z^2\overline{g'_\varepsilon(0,0)}+g'_\varepsilon(z,0)-g'_\varepsilon(0,0)\right)-f(z)g''_{z\varepsilon}(0,0)\right)+o(z,\varepsilon),\\ \varepsilon\in(0,\varepsilon_0), \tag{2}
\end{gather*}
where $\lim\limits_{\varepsilon\to+0}\frac{o(z,\varepsilon)}{\varepsilon}=0$ locally uniformly in
$E_z$;
\begin{equation}
f_3(z,\varepsilon)=f(z)+\varepsilon P_3(z)+o(z,\varepsilon),\quad \varepsilon\in(0,\hat\varepsilon), \tag{3}
\end{equation}
where
\begin{gather*}
P_3(z)=f'(z)(g'_\varepsilon(z,0)+z^2\overline{u}-u+itz)-\\
-f(z)(f''(0)(g'_\varepsilon(0,0)-u)+g''_{z\varepsilon}(0,0)+it)-g'_\varepsilon(0,0)+u,
\end{gather*}
$\hat\varepsilon=\min\left(\varepsilon_0,\frac1{|u|}\right)$,
$u$,
$t$ are constants,
$u\in\mathbb{C}$,
$t\in\mathbb{R}$, and $\lim\limits_{\varepsilon\to+0}\frac{o(z,\varepsilon)}{\varepsilon}=0$ locally uniformly in
$E_z$;
\begin{gather*}
f_4(z,\varepsilon)=f(z)+\varepsilon\left(f'(z)\left(z^2\overline{g'_\varepsilon(0,0)}+g'_\varepsilon(z,0)-g'_\varepsilon(0,0)+itz\right)-f(z)(g''_{z\varepsilon}(0,0)+it)\right)+\\
+o(z,\varepsilon),\quad
\varepsilon\in(0,\hat\varepsilon),\tag{4}
\end{gather*}
where
$t$ is a constant,
$t\in\mathbb{R}$, and $\lim\limits_{\varepsilon\to+0}\frac{o(z,\varepsilon)}{\varepsilon}=0$ locally uniformly in
$E_z$.
A number of new small variations have been obtained. In adition, the P. P. Kufarev method of
finding parameters in the Christoffel–Schwarz integral is illustrated by a simple example.
Keywords:
holomorphic univalent mapping, variational formula, parameters in the Christoffel–Schwarz integral, Kufarev method.
UDC:
517.54 Received: 12.07.2017
DOI:
10.17223/19988621/49/1
Fulltext:
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