MATHEMATICS

G. G. Shlyonskii

EXTREMAL PROBLEMS FOR DIFFERENTIABLE FUNCTIONALS IN THE THEORY OF UNIVALENT FUNCTIONS

(Presented by Academician V. I. Smirnov, 13 X 1956)

§ 1. Let \(G\) be some finitely connected domain, and let \(e\) be a closed set, \(e \subset G\).

Denote by \(\mathfrak{M}_G\) the family of functions regular in the domain \(G\), except possibly for a finite number of poles; by \(\mathfrak{R}_G\), the family of functions regular in \(G\); and by \(K_e\), some class of functions regular in \(G\), except possibly for a finite number of poles in \(e\).

Suppose that a certain real functional \(\Phi[f]\) is defined on \(K_e\). If there exists a function \(f_0 \in K_e\) such that \(\Phi[f] \leq \Phi[f_0]\) (maximum) or \(\Phi[f] \geq \Phi[f_0]\) (minimum) for all \(f \in K_e\), then the function \(f_0\) will be called an extremal function of the first kind for the functional \(\Phi[f]\) on \(K_e\).

Suppose that functionals \(\Phi_1[f], \ldots, \Phi_n[f]\) \((n \geq 1)\) (complex or real) are defined on \(K_e\). Consider in the \(n\)-dimensional vector space \(C_n\) (real or complex) the vector \(\vec{\Phi}[f]\), whose components are \(\Phi_1[f], \ldots, \Phi_n[f]\). We shall call \(\vec{\Phi}[f]\) a functional. Thus the functional \(\vec{\Phi}[f]\) is defined on \(K_e\). The set of elements of \(C_n\) which \(\vec{\Phi}[f]\) assumes on the whole class \(K_e\) will be called the range of variation of \(\vec{\Phi}[f]\) on \(K_e\) and denoted by \(\{\vec{\Phi}[f]\}_{K_e}\) or \(\{(\Phi_1[f], \ldots, \Phi_n[f])\}_{K_e}\). We introduce in \(C_n\) all the usual definitions of set theory. If there exists a function \(f_0 \in K_e\) such that \(\vec{\Phi}[f_0]\) is a boundary vector of the range of variation \(\{\vec{\Phi}[f]\}_{K_e}\), then the function \(f_0\) will be called a boundary function or an extremal function of the second kind*.

§ 2. Consider functionals defined on \(\mathfrak{M}_G\), taking finite values on \(K_e\), and weakly differentiable on \(K_e\), i.e., such that for any pair of functions \(f\) and \(h\) \((f \in K_e,\ h \in \mathfrak{M}_G)\) there exists a finite or infinite limit (with \(\lambda\) real)

\[ \lim_{\lambda \to 0} \frac{1}{\lambda}\{\Phi[f+\lambda h]-\Phi[f]\}; \]

this limit, depending on \(f\) and \(h\), is called the functional derivative.

We introduce the following definitions:

1. A real functional \(\Phi[f]\) is called a functional of type \(A_{K_e}\) if its functional derivative is equal to \(\operatorname{Re} D_f^{(\Phi)}[h]\), where \(D_f^{(\Phi)}[h]\) is a complex functional, distributive with respect to \(h \in \mathfrak{M}_G\), and assuming only finite values for \(h \in \mathfrak{R}_G\).

* In the usual terminology, generally speaking, \(\vec{\Phi}[f]\) is an operator.

1. A real functional of type \(A_{K_e}\) is called a functional of type \(A_{K_e}^l\) if \(D_f^{(\Phi)}[h]\) is continuous with respect to \(h\in \mathfrak A_G\)* and the function
   \[ \varphi(z)=D_f^{(\Phi)}\left[\frac{1}{\zeta-z}\right],\quad \zeta\in G, \]
   is a rational function having poles only in \(G\) of order \(\leq l\) and at least one pole of order \(l\).

3–4. A complex functional is called a functional of type \(B_{K_e}\) (respectively of type \(B_{K_e}^l\)) if its functional derivative is equal to \(D_f^{(\Phi)}[h]\), possessing the same properties as in 1 (respectively in 2).

§ 3. Let us give an important example of functionals of type \(B_{K_e}^l\). Let
\[ \Phi_k[f]=\sum_{\nu=1}^{p_k} a_{k\nu}^{(l_k)} f^{(l_k)}(z_{k\nu}),\quad z_{k\nu}\in G-e,\quad k=1,\ldots,n. \]
Then the functional
\[ \Phi[f]=F(\Phi_1[f],\ldots,\Phi_n[f]) \]
is a functional of type \(B_{K_e}^l\), if \(F(w_1,\ldots,w_n)\) is any function, defined for all values of its \(n\) complex variables, regular in an open domain containing \(\{\Phi[f]\}_{K_e}\), and having in this domain a nonvanishing gradient.

In this case
\[ D_f^{(\Phi)}[h]=\sum_{k=1}^{n}\alpha_k[f]\sum_{\nu=1}^{p_k}a_{k\nu}^{(l_k)}h^{(l_k)}(z_{k\nu}), \]
where
\[ \alpha_k[f]=\left.\frac{\partial}{\partial w_k}F(w_1,\ldots,w_n)\right|_{(w_1,\ldots,w_n)=(\Phi_1[f],\ldots,\Phi_n[f])},\quad k=1,\ldots,n. \]

§ 4. By the variational method (1) one can investigate the properties of extremal functions of both the first and the second kind for the functionals defined in § 2, on various classes of univalent functions.

Take, for example, as \(G\) the unit disk \(|z|<1\), and as \(K_e\) the class \(S\) of functions \(f(z)=z+c_2z^2+\cdots\), regular and univalent in \(|z|<1\).

Theorem 1. If \(f\) is an extremal function of the first kind for a functional \(\Phi[f]\) of type \(A_S\) on \(S\), then \(f\) satisfies the equation
\[ \left(\frac{zf'(z)}{f(z)}\right)^2 D_f^{(\Phi)}\left[\frac{f(\zeta)^2}{f(\zeta)-f(z)}\right] = \]
\[ =\frac{1}{2}D_f^{(\Phi)}\left[f(\zeta)+\zeta f'(\zeta)\frac{\zeta+z}{\zeta-z}\right] +\frac{1}{2}D_f^{(\Phi)}\left[f(\zeta)+\zeta f'(\zeta)\frac{\zeta+1/\bar z}{\zeta-1/\bar z}\right]. \tag{1} \]

The right-hand side of (1) is real on \(|z|=1\) and \(\leq 0\) (in the case of a maximum) or \(\geq 0\) (in the case of a minimum).

Theorem 2. If, under the hypotheses of Theorem 1, the functional \(\Phi[f]\) is of type \(A_S^l\) (\(l>2\) for the case when
\[ \varphi(z)=D_f^{(\Phi)}\left[\frac{1}{\zeta-z}\right] \]
has a pole of order \(l\) only at \(z=0\)), then \(f\) maps \(|z|<1\) onto the whole plane with slits along a finite number of analytic curves, which, under a suitable choice of the real parameter \(\tau\), are integral curves of the differential equation
\[ \left(\frac{dw}{d\tau}\right)^2 \frac{1}{w^2} D_f^{(\Phi)}\left[\frac{f(\zeta)^2}{f(\zeta)-w}\right] =\pm 1 \tag{2} \]
(the plus sign on the right-hand side of (2) corresponds to the case of a maximum).

* The definition of continuity of a functional on \(\mathfrak A_G\) is the usual one; convergence of elements \(h_n\in\mathfrak A_G,\ n=1,2,\ldots,\) is understood as uniform convergence of \(h_1,h_2,\ldots\) inside the domain \(G\).

§ 5. It is known (see, for example, (¹)) that to each function \(f(z)\in S\) mapping \(|z|<1\) onto the entire plane with a finite number of analytic slits one can associate a complex function \(K(t)\), \(|K(t)|=1\), continuous for all \(t\), \(0\le t<\infty\), except for a finite number of points \(t=t_1,\ldots,t_m\), such that
\[ f(z)=\lim_{t\to 0} e^t f(z,t), \]
where \(f(z,t)\) is the solution of the differential equation
\[ \frac{\partial f}{\partial t} = -f\,\frac{1+K(t)f}{1-K(t)f} \]
with the initial condition
\[ f\big|_{t=0}=z. \]

Theorem 3. Under the hypotheses of Theorem 2, the extremal function \(w=f(z)\) satisfies, for every \(t\), \(0\le t<\infty\), the equation
\[ \left(\frac{zf'(z)}{f(z)}\right)^2 \left(\frac{f(z,t)}{zf'(z,t)}\right)^2 D_f^{(\Phi)} \left[ \frac{f(\zeta)^2}{f(\zeta)-f(z)} \right] = \]
\[ = \frac12 D_f^{(\Phi)} \left[ f(\zeta)+ \frac{f(\zeta,t)}{f'(\zeta,t)}\,f'(\zeta) \frac{f(\zeta,t)+f(z,t)}{f(\zeta,t)-f(z,t)} \right] + \]
\[ + \frac12 D_f^{(\Phi)} \left[ f(\zeta)+ \frac{f(\zeta,t)}{f'(\zeta,t)}\,f'(\zeta) \frac{ f(\zeta,t)+\dfrac{1}{\overline{f(z,t)}} }{ f(\zeta,t)+\dfrac{1}{f(z,t)} } \right]. \tag{3} \]

For all \(t\), the right-hand side of (3) is real and \(\le 0\) (or, respectively, \(\ge 0\)) for such \(z\) that \(|f(z,t)|=1\).

Moreover, the following relations hold:
\[ \operatorname{Re} D_f^{(\Phi)} \left[ f(\zeta)- \frac{f(\zeta,t)}{f'(\zeta,t)}\,f'(\zeta) \frac{1+K(t)f(\zeta,t)}{1-K(t)f(\zeta,t)} \right] =0; \tag{4} \]
\[ \operatorname{Re} D_f^{(\Phi)} \left[ f(\zeta)- \frac{f(\zeta,t)}{f'(\zeta,t)}\,f'(\zeta) \frac{1+x f(\zeta,t)}{1-x f(\zeta,t)} \right] \begin{cases} \le 0\quad (\text{maximum}),\\ \ge 0\quad (\text{minimum}), \end{cases} \tag{5} \]
where \(0\le t<\infty\), \(x\) is any point of the disk \(|x|<1\). The function \(K(t)=e^{i\vartheta(t)}\) has derivatives of all orders with respect to \(t\), \(0\le t<\infty\), \(t\ne t_1,\ldots,t_m\). In addition,
\[ \frac{d\vartheta}{dt} = - \frac{ \operatorname{Im} D_f^{(\Phi)} \left[ \dfrac{f(\zeta,t)}{f'(\zeta,t)}\,f'(\zeta)\, \dfrac{2K(t)^2 f(\zeta,t)^2}{(1-K(t)f(\zeta,t))^4} \right] }{ \operatorname{Re} D_f^{(\Phi)} \left[ \dfrac{f(\zeta,t)}{f'(\zeta,t)}\,f'(\zeta)\, \dfrac{K(t)f(\zeta,t)(1+K(t)f(\zeta,t))}{(1-K(t)f(\zeta,t))^3} \right] }. \tag{6} \]

§ 6. Analogous results are obtained also for extremal functions of the second kind.

Leningrad State University
named after A. A. Zhdanov

Received
11 X 1956

REFERENCES

¹ G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, 1952.