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''method of contour integration''

An important method in the geometric theory of functions of a complex variable, enabling one to obtain various inequalities which express extremal properties of univalent and multivalent functions, as well as identities between the mapping functions of a domain (fundamental domain functions) in the theory of conformal mapping. The method makes essential use of properties of functions conformally mapping a given domain onto various canonical domains. By means of such mappings one can construct domain functions possessing the following boundary property: On each boundary component of the domain, the values of the function differ by an additive constant from the complex conjugate values corresponding to another such function. The method of boundary integration consists basically of the following.

Some integral is considered, taken over the entire boundary of a given domain (the boundary can generally be taken to consist of a finite number of simple closed analytic curves). This integral is chosen so that the integrand contains factors with the boundary property indicated above and such that, after using this property, an integral is obtained that can be calculated by means of the residue theorem (cf. [[Contour integration, method of|Contour integration, method of]]; [[Cauchy integral theorem|Cauchy integral theorem]]). If from other considerations either the value of the original integral is known or its sign, then one obtains as a result some relation between the functions used or some inequality connecting them. Often the boundary integral to which the above method can be applied appears as the result of a transformation according to Green's formula of a non-negative double integral, namely, the integral of the square of the modulus of the derivative of some function that is regular in the given domain. Hence the connection between the method of boundary integration and the [[Area method|area method]]. By means of the method of boundary integration, results have been obtained concerning: [[Distortion theorems|distortion theorems]] for univalent conformal mappings between multiply-connected domains (see [[#References|[1]]], [[#References|[2]]]); necessary and sufficient conditions on the coefficients of univalent functions (see [[#References|[3]]]); and identities relating the fundamental domain functions in the theory of conformal mapping (see [[#References|[4]]]).

The method of boundary integration is also employed in the study of univalent functions in the following form. Suppose, for example, that  $  B $
is a domain in the  $  w $-
plane with boundary  $  C $
consisting of a finite number of simple closed analytic curves; suppose that  $  S ( w) $
is a function that is harmonic in the entire  $  w $-
plane except for a finite number of points of  $  B $; 
and suppose that  $  p ( w) $
is a function with the following property: The difference  $  S ( w) - p ( w) $
is harmonic in the domain  $  B $, 
continuous in the closed domain, and  $  p ( w) \mid  _ {C} = 0 $. 
Then

$$ 
\int\limits _ { C } S 
\frac{\partial  p }{\partial  n }
  d s  \leq   0 ,
$$

where  $  \partial  / \partial  n $
denotes differentiation along the outward normal to  $  B $. 
If  $  \sigma ( w) $
and  $  q( w) $
are analytic functions for which  $  S =  \mathop{\rm Re}  \sigma $, 
$  p =  \mathop{\rm Re}  q $, 
then the above inequality can be written in the form

$$ 
 \mathop{\rm Re} \left \{ 
\frac{1}{i}
 \int\limits _ { C }
( \sigma - q ) \sigma  ^  \prime   d w \right \}  \leq   0 .
$$

The integral in this inequality can be calculated by the residue theorem. By choosing various functions  $  S ( w) $
and  $  p ( w) $
that are suitably related to the functions under investigation, one can thus obtain various new inequalities for univalent functions (see [[#References|[5]]]–[[#References|[7]]]).

The method of boundary integration is also successfully employed in the study of non-univalent conformal mappings. Thus, there have been established by this method a number of new extremal properties of functions that are meromorphic in a multiply-connected domain and satisfy certain additional properties (see [[#References|[8]]]); and generalizations have been obtained for multiply-connected domains, for the case of several poles and for functions that are  $  p $-
valent in an appropriate generalized sense, of Goluzin's area theorem for functions that are  $  p $-
valent in the disc (see [[#References|[9]]]). The domain functions referred to above are closely related to Bergman kernel functions (cf. [[Bergman kernel function|Bergman kernel function]]), and results obtained by the method of boundary integration are often expressed in terms of them. Hence there is also a connection between the method of boundary integration and the theory of orthonormal systems of analytic functions.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Grunsky,   "Neue Abschätzungen zur konformen Abbildung ein- und mehrfachzusammenhängender Bereiche"  ''Schriften Math. Sem. Inst. Angew. Math. Univ. Berlin''  (1932)  pp. 95–140</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,   "On distortion theorems for conformal mappings of multiply-connected domains"  ''Mat. Sb.'' , '''2''' :  1  (1937)  pp. 37–63  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Grunsky,   "Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen"  ''Math. Z.'' , '''45'''  (1939)  pp. 29–61</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.R. Garabedian,   M. Schiffer,   "Identities in the theory of conformal mapping"  ''Trans. Amer. Math. Soc.'' , '''65''' :  2  (1949)  pp. 187–238</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  Z. Nehari,   "Some inequalities in the theory of functions"  ''Trans. Amer. Math. Soc.'' , '''75''' :  2  (1953)  pp. 256–286</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  Yu.E. Alenitsyn,   "On univalent functions in multiply-connected domains"  ''Mat. Sb.'' , '''39''' :  3  (1956)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  Yu.E. [Yu.E. Alenitsyn] Alenicyn,   "Univalent functions without common values in a multiply connected domain"  ''Proc. Steklov Inst. Math.'' , '''94'''  (1968)  pp. 1–18  ''Trudy Mat. Inst. Steklov.'' , '''94'''  (1968)  pp. 4–18</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  H. Meschkowski,   "Beiträge zur Theorie der Orthonomalsysteme"  ''Mat. Ann.'' , '''127'''  (1954)  pp. 107–129</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  Yu.E. [Yu.E. Alenitsyn] Alenicyn,   "Area theorems for functions analytic in a finitely connected domain"  ''Math. USSR Izv.'' , '''7''' :  5  (1973)  pp. 1130–1151  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''37''' :  5  (1973)  pp. 1132–1154</TD></TR><TR><TD valign="top">[10]</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></table>

====Comments====
The mapping functions of a domain, or fundamental domain functions, are univalent conformal mappings of the given domain onto certain standard domains, e.g. from a simply-connected domain onto the unit disc (cf. [[Riemann theorem|Riemann theorem]]) or from a doubly-connected domain onto a standard annulus, [[#References|[10]]].