On small variation formulas

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.