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])}), then the function (f_0) will be called a }). 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_eboundary 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.