On small variation formulas

Ya. V. Borisova ; I. A. Kolesnikov ; S. A. Kopanev

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 49 (2017), pp. 5-15

Résumé

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.

Détail

Keywords: holomorphic univalent mapping, parameters in the Christoffel–Schwarz integral, Kufarev method.
Mots-clés : variational formula