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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p0715101.png" /></td> </tr></table>

with given real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p0715102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p0715103.png" />, and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p0715104.png" /> (the kernel of the class), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p0715105.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p0715106.png" /> is the class of non-decreasing functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p0715107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p0715108.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p0715109.png" /> 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151010.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151011.png" />, and the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151012.png" /> is known (see [[#References|[1]]], Chapt. 11, [[#References|[3]]]).

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

1) The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151013.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151014.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151015.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151016.png" /> admitting a representation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151017.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151018.png" /> are fixed continuous real-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151020.png" /> coincides with the closed convex hull <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151021.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151022.png" /> of the points

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151023.png" /></td> </tr></table>

(a theorem of Riesz).

2) Every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151024.png" /> can be represented in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151025.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151029.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151030.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151031.png" /> (a theorem of Carathéodory).

3) There exists at least one non-decreasing function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151033.png" />, such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151034.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151035.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151037.png" /> are given real-valued continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151039.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151040.png" /> are given complex numbers, if and only if whenever the complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151041.png" /> satisfy

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151042.png" /></td> </tr></table>

then also

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071510/p07151043.png" /></td> </tr></table>

(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 [[#References|[1]]], Appendix; [[#References|[4]]], ).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5c]</TD> <TD valign="top">  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</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,   "Univalent functions" , Springer  (1983)  pp. Sect. 10.11</TD></TR></table>