Parametric integral-representation method

A method in the geometric theory of functions of a complex variable that is used to solve extremal problems in function classes by representing these classes using integrals depending on parameters.

Among these classes are the Carathéodory class, the class of univalent star-like functions in the disc, and the class of typically-real functions (cf. Star-like function and Typically-real function). The functions of these classes have parametric representations involving a Stieltjes integral

$$ \int\limits _ { a } ^ { b } g( z, t) d \mu ( t) $$

with given real numbers $ a $ and $ b $, and a function $ g( z, t) $ (the kernel of the class), $ \mu ( t) \in M _ {a,b} $, where $ M _ {a,b} $ is the class of non-decreasing functions on $ [ a, b] $, $ \mu ( b) - \mu ( a) = 1 $ ($ \mu $ is the parameter of the class).

For classes of functions having a parametric representation by Stieltjes integrals, variational formulas have been obtained that show, in the solution of extremal problems in these classes, that the extremal function is of the form

$$ f( z) = \sum _ { k= 1} ^ { m } \lambda _ {k} g( z, t _ {k} ),\ \ \lambda _ {k} \geq 0,\ \ \sum _ { k= 1} ^ { m } \lambda _ {k} = 1, $$

where $ t _ {k} \in [ a, b] $, and the value of $ m $ is known (see [1], Chapt. 11, [3]).

To find the ranges of functionals and systems of functionals on such classes the following theorems are sometimes useful.

1) The set $ B $ of points $ x = ( x _ {1}, \dots, x _ {n} ) $ of the $ n $-dimensional Euclidean space $ \mathbf R ^ {n} $ admitting a representation

$$ x _ {k} = \int\limits _ { a } ^ { b } u _ {k} ( t) d \mu ( t),\ \ k = 1, \dots, n, $$

where the $ u _ {k} ( t) $ are fixed continuous real-valued functions on $ [ a, b] $ and $ \mu ( t) \in M _ {a,b} $ coincides with the closed convex hull $ R( U) $ of the set $ U $ of the points

$$ x _ {k} = u _ {k} ( t) ,\ \ k = 1, \dots, n,\ \ a \leq t \leq b $$

(a theorem of Riesz).

2) Every point $ x = ( x _ {1}, \dots, x _ {n} ) \in R( U) \subset \mathbf R ^ {n} $ can be represented in the form

$$ x _ {k} = \sum _ { j= 1} ^ { m } \lambda _ {j} u _ {k} ( t _ {j} ),\ \ k = 1, \dots, n, $$

where $ \lambda _ {j} > 0 $, $ j = 1, \dots, m $, $ \sum _ {j= 1} ^ {m} \lambda _ {j} = 1 $, $ m \leq n+ 1 $, and if $ x \in \partial R( U) $, then $ m \leq n $ (a theorem of Carathéodory).

3) There exists at least one non-decreasing function $ \mu ( t) $, $ a \leq t \leq b $, such that

$$ \int\limits _ { a } ^ { b } w _ {k} ( t) d \mu ( t) = \gamma _ {k} ,\ \ k = 1, \dots, n, $$

where

$$ w _ {1} ( t) \equiv 1 ,\ \ w _ {k} ( t) = u _ {k} ( t) + iv _ {k} ( t),\ \ k = 1, \dots, n, $$

$ u _ {k} ( t) $, $ v _ {k} ( t) $ are given real-valued continuous functions on $ [ a, b] $, $ \gamma _ {1} > 0 $, and $ \gamma _ {k} $ are given complex numbers, if and only if whenever the complex numbers $ \alpha _ {1}, \dots, \alpha _ {n} $ satisfy

$$ \sum _ { k= 1} ^ { n } [ \alpha _ {k} w _ {k} ( t) + \overline \alpha _ {k} \overline{w} _ {k} ( t)] \geq 0,\ \ a \leq t \leq b, $$

then also

$$ \sum _ { k= 1} ^ { n } [ \alpha _ {k} \gamma _ {k} + \overline \alpha _ {k} \overline \gamma _ {k} ] \geq 0 $$

(a theorem of Riesz).

These theorems make it possible to give geometric and algebraic characterizations of the ranges of systems of coefficients and individual coefficients on classes of functions that are regular and have positive real part in the disc (or an annulus), or are regular and typically real in the disc (or annulus), and on some other classes (see [1], Appendix; [4], ).

References

[1]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[2]	M.G. Krein, "The ideas of P.L. Chebyshev and A.A. Markov in the theory of limiting values of integrals, and their further development" Uspekhi Mat. Nauk , 6 : 4 (1951) pp. 3–120 (In Russian)
[3]	N.A. Lebedev, I.A. Aleksandrov, "On the variational method in classes of functions representable by means of Stieltjes integrals" Proc. Steklov Inst. Math. , 94 (1968) pp. 91–103 Trudy Mat. Inst. Steklov. , 94 (1968) pp. 79–89
[4]	E.G. Goluzina, "The value domains of the coefficient systems of a certain class of functions meromorphic in a disk" Proc. Steklov Inst. Math. , 94 (1968) pp. 37–52 Trudy Mat. Inst. Steklov. , 94 (1968) pp. 33–46
[5a]	E.G. Goluzina, "On domains of values of systems of functionals in some classes of functions, representable by a Stieltjes integral" J. Soviet Math. , 2 : 6 (1974) pp. 582–605 Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. , 24 (1972) pp. 29–62
[5b]	E.G. Goluzina, "Ranges of values of systems of coefficients in the class of functions with positive real part in an annulus" J. Soviet Math. , 8 : 6 (1977) pp. 642–661 Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. , 44 (1972) pp. 17–40
[5c]	E.G. Goluzina, "Ranges of certain systems of functionals in classes of functions with a positive real part" J. Soviet Math. , 19 : 6 (1982) pp. 1630–1636 Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. , 100 (1980) pp. 17–25

References

[a1]	P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11