View source for Variation-parametric method

A method which is a synthesis of Goluzin's method of variations (cf. [[Internal variations, method of|Internal variations, method of]]) and Loewner's [[Parametric representation method|parametric representation method]] for the important subclass of univalent functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v0961801.png" /> mapping the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v0961802.png" /> onto domains obtained from the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v0961803.png" /> by cutting along piecewise-continuous arcs. This synthesis is obtained by a special variation which, in the simplest case of one Jordan cut, is determined by the following theorem. Let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v0961804.png" /> map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v0961805.png" /> onto the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v0961806.png" /> obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v0961807.png" /> by performing the cut

<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/v/v096/v096180/v0961808.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v0961809.png" /> is continuous, while the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618011.png" />, is simply connected. A parametrization of the cut <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618012.png" /> may be considered such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618014.png" />, associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618015.png" />, which univalently and conformally maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618016.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618017.png" />, is normalized by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618018.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618019.png" /> denote the function inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618020.png" /> for a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618021.png" />. Then, for all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618022.png" />, (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618023.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618024.png" />) and for all constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618025.png" />, there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618027.png" /> which may be represented in the form

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Here

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and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618033.png" /> is a holomorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618034.png" />, the limit of which with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618035.png" /> uniformly tends to zero inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618036.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618037.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618038.png" />).

If, in the course of study of extremal problems in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618039.png" />, use is made of the special variation mentioned above and of Loewner's equation

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which is satisfied by a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618042.png" /> subject to the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618043.png" />, two equations are usually obtained for the function which was associated to the extremal function. Irrespective of the constants contained in the equations, which can be expressed as values of the extremal function, further study of these equations frequently yields complete solutions of the problems under consideration, in particular in the problem of the domain of values of a functional which depends analytically on the function, its derivative and their conjugate values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618044.png" />. The method was proposed by P.P. Kufarev [[#References|[1]]]; for subsequent development and applications of the method see [[#References|[2]]]–[[#References|[5]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.P. Kufarev,   "On one property of extremal domains of coefficient problems"  ''Dokl. Akad. Nauk SSSR'' , '''97''' :  3  (1954)  pp. 391–393  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.A. Aleksandrov,   ''Uchen. Zap. Tomsk. Univ.'' , '''32'''  (1958)  pp. 41–57</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.A. Aleksandrov,   "Boundary values of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618045.png" /> on the class of holomorphic univalent functions in the disc"  ''Sibirsk. Mat. Zh.'' , '''4''' :  1  (1963)  pp. 17–31  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.I. Red'kov,   "The domains of values of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618046.png" /> for certain classes of bounded univalent functions"  ''Soviet Math. Dokl.'' , '''1''' :  2  (1960)  pp. 848–851  ''Dokl. Akad. Nauk SSSR'' , '''133''' :  2  (1960)  pp. 284–287</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.I. Red'kov,   "The domain of values of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618047.png" /> on the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618048.png" />"  ''Izv. Vyssh. Uchebn. Zaved. Mat.'' , '''4(29)'''  (1962)  pp. 134–142  (In Russian)</TD></TR></table>



====Comments====
The idea to combine variational methods and Loewner theory seems to go back to M. Schiffer, cf. [[#References|[a2]]], Chapts. 10, 11; see [[#References|[a2]]], also for further references.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.A. Aleksandrov,   "Parametric extensions in the theory of univalent functions" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.L. Duren,   "Univalent functions" , Springer  (1983)  pp. Sect. 10.11</TD></TR></table>