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''general coefficient theorem''
 
''general coefficient theorem''
  
 
A theorem in the theory of univalent conformal mappings of families of domains on a Riemann surface, containing an inequality for the coefficients of the mapping functions, as well as conditions to be satisfied by the function so that the inequality becomes an equality. Jenkins' theorem is an exact expression and generalization of Teichmüller's principle (stated without proof, [[#References|[1]]]), according to which the solution of a certain class of extremal problems for univalent functions is determined by the quadratic differentials of the respective forms. Obtained by J.A. Jenkins in 1954 [[#References|[1]]]–[[#References|[4]]].
 
A theorem in the theory of univalent conformal mappings of families of domains on a Riemann surface, containing an inequality for the coefficients of the mapping functions, as well as conditions to be satisfied by the function so that the inequality becomes an equality. Jenkins' theorem is an exact expression and generalization of Teichmüller's principle (stated without proof, [[#References|[1]]]), according to which the solution of a certain class of extremal problems for univalent functions is determined by the quadratic differentials of the respective forms. Obtained by J.A. Jenkins in 1954 [[#References|[1]]]–[[#References|[4]]].
  
The conditions of Jenkins' theorem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j0542001.png" /> be a finite oriented [[Riemann surface|Riemann surface]], let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j0542002.png" /> be a positive [[Quadratic differential|quadratic differential]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j0542003.png" /> with at least one pole of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j0542004.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j0542005.png" /> be all the poles of order 2, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j0542006.png" /> are all the poles of orders higher than 2. Let an open and everywhere-dense set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j0542007.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j0542008.png" /> be the complement of the union of a finite number of closures of trajectories and closures of trajectory arcs, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j0542009.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420010.png" />. Suppose the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420011.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420012.png" /> conformally and univalently (cf. [[Conformal mapping|Conformal mapping]]) into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420013.png" />, and suppose there exists a homotopy
+
The conditions of Jenkins' theorem. Let $  {\mathcal R} $
 +
be a finite oriented [[Riemann surface|Riemann surface]], let $  Q ( z)  dz  ^ {2} $
 +
be a positive [[Quadratic differential|quadratic differential]] on $  {\mathcal R} $
 +
with at least one pole of order $  \geq  2 $,  
 +
and let $  P _ {1} \dots P _ {r} $
 +
be all the poles of order 2, while $  P _ {r+1} \dots P _ {p} $
 +
are all the poles of orders higher than 2. Let an open and everywhere-dense set $  \Delta $
 +
on $  {\mathcal R} $
 +
be the complement of the union of a finite number of closures of trajectories and closures of trajectory arcs, and let $  P _ {j} \in \Delta $,  
 +
$  j = 1 \dots p $.  
 +
Suppose the function $  f _ {0} ( P) $
 +
maps $  \Delta $
 +
conformally and univalently (cf. [[Conformal mapping|Conformal mapping]]) into $  {\mathcal R} $,
 +
and suppose there exists a homotopy
 +
 
 +
$$
 +
f _ {t} ( P) :  ( \Delta \times [ 0 , 1 ] )  \rightarrow  {\mathcal R}
 +
$$
 +
 
 +
of the mapping  $  f _ {0} ( P) $
 +
into the identity mapping  $  f _ {1} ( P) \equiv P $
 +
which leaves all poles from  $  \Delta $
 +
fixed and satisfies the condition  $  f _ {t} ( P) \neq R $
 +
for each pole  $  R \in {\mathcal R} $,
 +
$  t \in [ 0 , 1 ] $,
 +
and each point  $  P \neq R $.
 +
Let  $  z _ {j} = z _ {j} ( P) $
 +
be a local parameter for the pole  $  P _ {j} $
 +
such that  $  z _ {j} ( P _ {j} ) = \infty $,
 +
$  j = 1 \dots p $.  
 +
Let, for  $  j = 1 \dots p $,  
 +
in a neighbourhood of the pole  $  P _ {j} $,
 +
 
 +
$$
 +
Q ( z _ {j} )  =  \left \{
 +
 
 +
\begin{array}{ll}
 +
\alpha  ^ {(j)} z _ {j}  ^ {-2} +
 +
\textrm{ higher  powers  of  }  z _ {j}  ^ {-1}  & \textrm{ if }  j \leq  r,  \\
 +
\alpha  ^ {(j)} \left [ z _ {j} ^ {m _ {j} - 4 } + \sum _ {s =
 +
1 } ^  \infty  \beta _ {s}  ^ {(j)} z _ {j} ^ {m _ {j} - s - 4 } \right ]  & \textrm{ if }  j > r ;  \\
 +
\end{array}
 +
 
 +
$$
  
<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/j/j054/j054200/j05420014.png" /></td> </tr></table>
+
$$
 +
f _ {0} ( z _ {j} )  = \left \{
 +
\begin{array}{ll}
 +
\alpha  ^ {(j)} z _ {j} + \textrm{ non- positive  powers  of  }  z _ {j}  & \textrm{ if }  j \leq  r ,  \\
 +
z _ {j} +
 +
\sum _ {s = n _ {j} } ^  \infty  a _ {s}  ^ {(j)} z _ {j}  ^ {-s}  & \textrm{ if }  j> r ,  \\
 +
\end{array}
  
of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420015.png" /> into the identity mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420016.png" /> which leaves all poles from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420017.png" /> fixed and satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420018.png" /> for each pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420020.png" />, and each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420021.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420022.png" /> be a local parameter for the pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420023.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420025.png" />. Let, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420026.png" />, in a neighbourhood of the pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420027.png" />,
+
\right .$$
  
<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/j/j054/j054200/j05420028.png" /></td> </tr></table>
+
where  $  n _ {j} $
 +
is the integer part of the number  $  ( m _ {j} - 3)/2 $.  
 +
Let
  
<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/j/j054/j054200/j05420029.png" /></td> </tr></table>
+
$$
 +
\left . d ( P _ {j} )  = \lim\limits _ {P \rightarrow P _ {j} } \
 +
\mathop{\rm Arg}  z _ {j} [ f _ {t} ( P) ] \right | _ {t = 0 }  ^ {t = 1 } ,\  j = 1 \dots p ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420030.png" /> is the integer part of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420031.png" />. Let
+
and let  $  d ( P _ {j} ) = 0 $
 +
for all  $  j > r $
 +
for which  $  P _ {j} $
 +
lies on the boundary of a strip-like domain with respect to  $  Q ( z)  dz  ^ {2} $.  
 +
Finally, let
  
<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/j/j054/j054200/j05420032.png" /></td> </tr></table>
+
$$
 +
T _ {j}  = \alpha  ^ {(j)} \left [ a _ {m _ {j}  - 3 }  ^ {(j)} +
 +
\sum _ { s = 1 } ^ { {n _ j } } \beta _ {s}  ^ {(j)}
 +
a _ {m _ {j}  - s - 3 }  ^ {(j)} \right ] +
 +
$$
  
and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420033.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420034.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420035.png" /> lies on the boundary of a strip-like domain with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420036.png" />. Finally, let
+
$$
 +
+
 +
\left \{
 +
\begin{array}{ll}
 +
0 & \textrm{ for odd  }  m _ {j} ,  \\
 +
\alpha  ^ {(j)} \left (
  
<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/j/j054/j054200/j05420037.png" /></td> </tr></table>
+
\frac{m _ {j} }{4}
 +
- 1 \right ) \left ( a _ {n _ {j}  }  ^ {(j)}
 +
\right )  ^ {2}  & \textrm{ for  even  }  m _ {j} . \\
 +
\end{array}
  
<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/j/j054/j054200/j05420038.png" /></td> </tr></table>
+
$$
  
 
The statement of Jenkins' theorem. Under the conditions mentioned above,
 
The statement of Jenkins' theorem. Under the conditions mentioned above,
  
<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/j/j054/j054200/j05420039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\mathop{\rm Re} \left \{ \sum _ {j = 1 } ^ { r }  \alpha  ^ {(j)}
 +
\mathop{\rm ln}  a  ^ {(j)} + \sum _ {j = r + 1 } ^ { p }
 +
\alpha  ^ {(j)} T _ {j} \right \}  \leq  0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420041.png" />.
+
where $  \mathop{\rm ln}  a  ^ {(j)}  =   \mathop{\rm ln}  | a  ^ {(j)} | - i d ( P _ {j} ) $,  
 +
j \leq  r $.
  
Jenkins' theorem in the case of equality. If in (*) the equality sign holds, then: a) in each domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420042.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420043.png" /> is an isometry in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420044.png" />-metric: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420045.png" />, each trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420046.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420047.png" /> is mapped to a trajectory, and the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420048.png" /> is everywhere-dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420049.png" />; b) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420050.png" /> to be the identity mapping in a certain domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420051.png" /> it is sufficient that one of the following additional conditions holds:
+
Jenkins' theorem in the case of equality. If in (*) the equality sign holds, then: a) in each domain $  \Delta _ {l} \subset  \Delta $
 +
the mapping $  f _ {0} $
 +
is an isometry in the $  Q $-
 +
metric: $  | d \zeta | = | Q ( z)  |  ^ {1/2} |  dz | $,  
 +
each trajectory $  Q ( z)  dz  ^ {2} $
 +
in $  \Delta $
 +
is mapped to a trajectory, and the set $  f _ {0} ( \Delta ) $
 +
is everywhere-dense in $  {\mathcal R} $;  
 +
b) for $  f _ {0} $
 +
to be the identity mapping in a certain domain $  \Delta _ {l} \subset  \Delta $
 +
it is sufficient that one of the following additional conditions holds:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420052.png" /> contains a pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420054.png" />, of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420056.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420057.png" />;
+
1) $  \Delta _ {l} $
 +
contains a pole $  P _ {j} $,  
 +
j > r $,  
 +
of order $  m _ {j} $
 +
such that $  a _ {s}  ^ {(j)} = 0 $
 +
for $  s < \min ( n _ {j} + 1 , m _ {j} - 3 ) $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420058.png" /> contains a pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420059.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420061.png" />;
+
2) $  \Delta _ {1} $
 +
contains a pole $  P _ {j} $
 +
for which j \leq  r $
 +
and $  a  ^ {(j)} = 1 $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420062.png" /> contains a point of a trajectory ending in a simple pole.
+
3) $  \Delta _ {1} $
 +
contains a point of a trajectory ending in a simple pole.
  
If (*) is an equality and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420063.png" /> for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420064.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420065.png" /> is conformally equivalent to the sphere, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420066.png" /> has no zeros or simple poles and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420067.png" />. If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420068.png" /> is a domain, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054200/j05420069.png" /> is conformally equivalent to a linear mapping all fixed points of which are images of the poles.
+
If (*) is an equality and if $  | a  ^ {(j)} | \neq 1 $
 +
for a certain j \leq  r $,  
 +
then $  {\mathcal R} $
 +
is conformally equivalent to the sphere, $  Q ( z)  dz  ^ {2} $
 +
has no zeros or simple poles and $  r = p= 2 $.  
 +
If, in addition, $  \Delta $
 +
is a domain, the mapping $  f _ {0} $
 +
is conformally equivalent to a linear mapping all fixed points of which are images of the poles.
  
The method of the extremal metric (cf. [[Extremal metric, method of the|Extremal metric, method of the]]), on which the proof of Jenkins' theorem is based, was employed by Jenkins, with suitable modifications, to obtain several other results, in particular the so-called special coefficient theorem [[#References|[4]]]. For additions to and the development of Jenkins' theorem, see [[#References|[5]]].
+
The method of the extremal metric (cf. [[Extremal metric, method of the]]), on which the proof of Jenkins' theorem is based, was employed by Jenkins, with suitable modifications, to obtain several other results, in particular the so-called special coefficient theorem [[#References|[4]]]. For additions to and the development of Jenkins' theorem, see [[#References|[5]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Jenkins,  "An extension of the general coefficient theorem"  ''Trans. Amer. Math. Soc.'' , '''95''' :  3  (1960)  pp. 387–407</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Jenkins,  "The general coefficient theorem and certain applications"  ''Bull. Amer. Math. Soc.'' , '''68''' :  1  (1962)  pp. 1–9</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.A. Jenkins,  "Some area theorems and a special coefficient theorem"  ''Illinois J. Math.'' , '''8''' :  1  (1964)  pp. 80–99</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.M. Tamrazov,  "On the general coefficient theorem"  ''Math. USSR Sb.'' , '''1'''  (1967)  pp. 49–59  ''Mat. Sb.'' , '''72''' :  1  (1967)  pp. 59–71</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Jenkins,  "An extension of the general coefficient theorem"  ''Trans. Amer. Math. Soc.'' , '''95''' :  3  (1960)  pp. 387–407</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Jenkins,  "The general coefficient theorem and certain applications"  ''Bull. Amer. Math. Soc.'' , '''68''' :  1  (1962)  pp. 1–9</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.A. Jenkins,  "Some area theorems and a special coefficient theorem"  ''Illinois J. Math.'' , '''8''' :  1  (1964)  pp. 80–99</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.M. Tamrazov,  "On the general coefficient theorem"  ''Math. USSR Sb.'' , '''1'''  (1967)  pp. 49–59  ''Mat. Sb.'' , '''72''' :  1  (1967)  pp. 59–71</TD></TR></table>

Latest revision as of 17:42, 13 January 2024


general coefficient theorem

A theorem in the theory of univalent conformal mappings of families of domains on a Riemann surface, containing an inequality for the coefficients of the mapping functions, as well as conditions to be satisfied by the function so that the inequality becomes an equality. Jenkins' theorem is an exact expression and generalization of Teichmüller's principle (stated without proof, [1]), according to which the solution of a certain class of extremal problems for univalent functions is determined by the quadratic differentials of the respective forms. Obtained by J.A. Jenkins in 1954 [1][4].

The conditions of Jenkins' theorem. Let $ {\mathcal R} $ be a finite oriented Riemann surface, let $ Q ( z) dz ^ {2} $ be a positive quadratic differential on $ {\mathcal R} $ with at least one pole of order $ \geq 2 $, and let $ P _ {1} \dots P _ {r} $ be all the poles of order 2, while $ P _ {r+1} \dots P _ {p} $ are all the poles of orders higher than 2. Let an open and everywhere-dense set $ \Delta $ on $ {\mathcal R} $ be the complement of the union of a finite number of closures of trajectories and closures of trajectory arcs, and let $ P _ {j} \in \Delta $, $ j = 1 \dots p $. Suppose the function $ f _ {0} ( P) $ maps $ \Delta $ conformally and univalently (cf. Conformal mapping) into $ {\mathcal R} $, and suppose there exists a homotopy

$$ f _ {t} ( P) : ( \Delta \times [ 0 , 1 ] ) \rightarrow {\mathcal R} $$

of the mapping $ f _ {0} ( P) $ into the identity mapping $ f _ {1} ( P) \equiv P $ which leaves all poles from $ \Delta $ fixed and satisfies the condition $ f _ {t} ( P) \neq R $ for each pole $ R \in {\mathcal R} $, $ t \in [ 0 , 1 ] $, and each point $ P \neq R $. Let $ z _ {j} = z _ {j} ( P) $ be a local parameter for the pole $ P _ {j} $ such that $ z _ {j} ( P _ {j} ) = \infty $, $ j = 1 \dots p $. Let, for $ j = 1 \dots p $, in a neighbourhood of the pole $ P _ {j} $,

$$ Q ( z _ {j} ) = \left \{ \begin{array}{ll} \alpha ^ {(j)} z _ {j} ^ {-2} + \textrm{ higher powers of } z _ {j} ^ {-1} & \textrm{ if } j \leq r, \\ \alpha ^ {(j)} \left [ z _ {j} ^ {m _ {j} - 4 } + \sum _ {s = 1 } ^ \infty \beta _ {s} ^ {(j)} z _ {j} ^ {m _ {j} - s - 4 } \right ] & \textrm{ if } j > r ; \\ \end{array} $$

$$ f _ {0} ( z _ {j} ) = \left \{ \begin{array}{ll} \alpha ^ {(j)} z _ {j} + \textrm{ non- positive powers of } z _ {j} & \textrm{ if } j \leq r , \\ z _ {j} + \sum _ {s = n _ {j} } ^ \infty a _ {s} ^ {(j)} z _ {j} ^ {-s} & \textrm{ if } j> r , \\ \end{array} \right .$$

where $ n _ {j} $ is the integer part of the number $ ( m _ {j} - 3)/2 $. Let

$$ \left . d ( P _ {j} ) = \lim\limits _ {P \rightarrow P _ {j} } \ \mathop{\rm Arg} z _ {j} [ f _ {t} ( P) ] \right | _ {t = 0 } ^ {t = 1 } ,\ j = 1 \dots p , $$

and let $ d ( P _ {j} ) = 0 $ for all $ j > r $ for which $ P _ {j} $ lies on the boundary of a strip-like domain with respect to $ Q ( z) dz ^ {2} $. Finally, let

$$ T _ {j} = \alpha ^ {(j)} \left [ a _ {m _ {j} - 3 } ^ {(j)} + \sum _ { s = 1 } ^ { {n _ j } } \beta _ {s} ^ {(j)} a _ {m _ {j} - s - 3 } ^ {(j)} \right ] + $$

$$ + \left \{ \begin{array}{ll} 0 & \textrm{ for odd } m _ {j} , \\ \alpha ^ {(j)} \left ( \frac{m _ {j} }{4} - 1 \right ) \left ( a _ {n _ {j} } ^ {(j)} \right ) ^ {2} & \textrm{ for even } m _ {j} . \\ \end{array} $$

The statement of Jenkins' theorem. Under the conditions mentioned above,

$$ \tag{* } \mathop{\rm Re} \left \{ \sum _ {j = 1 } ^ { r } \alpha ^ {(j)} \mathop{\rm ln} a ^ {(j)} + \sum _ {j = r + 1 } ^ { p } \alpha ^ {(j)} T _ {j} \right \} \leq 0 , $$

where $ \mathop{\rm ln} a ^ {(j)} = \mathop{\rm ln} | a ^ {(j)} | - i d ( P _ {j} ) $, $ j \leq r $.

Jenkins' theorem in the case of equality. If in (*) the equality sign holds, then: a) in each domain $ \Delta _ {l} \subset \Delta $ the mapping $ f _ {0} $ is an isometry in the $ Q $- metric: $ | d \zeta | = | Q ( z) | ^ {1/2} | dz | $, each trajectory $ Q ( z) dz ^ {2} $ in $ \Delta $ is mapped to a trajectory, and the set $ f _ {0} ( \Delta ) $ is everywhere-dense in $ {\mathcal R} $; b) for $ f _ {0} $ to be the identity mapping in a certain domain $ \Delta _ {l} \subset \Delta $ it is sufficient that one of the following additional conditions holds:

1) $ \Delta _ {l} $ contains a pole $ P _ {j} $, $ j > r $, of order $ m _ {j} $ such that $ a _ {s} ^ {(j)} = 0 $ for $ s < \min ( n _ {j} + 1 , m _ {j} - 3 ) $;

2) $ \Delta _ {1} $ contains a pole $ P _ {j} $ for which $ j \leq r $ and $ a ^ {(j)} = 1 $;

3) $ \Delta _ {1} $ contains a point of a trajectory ending in a simple pole.

If (*) is an equality and if $ | a ^ {(j)} | \neq 1 $ for a certain $ j \leq r $, then $ {\mathcal R} $ is conformally equivalent to the sphere, $ Q ( z) dz ^ {2} $ has no zeros or simple poles and $ r = p= 2 $. If, in addition, $ \Delta $ is a domain, the mapping $ f _ {0} $ is conformally equivalent to a linear mapping all fixed points of which are images of the poles.

The method of the extremal metric (cf. Extremal metric, method of the), on which the proof of Jenkins' theorem is based, was employed by Jenkins, with suitable modifications, to obtain several other results, in particular the so-called special coefficient theorem [4]. For additions to and the development of Jenkins' theorem, see [5].

References

[1] J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958)
[2] J.A. Jenkins, "An extension of the general coefficient theorem" Trans. Amer. Math. Soc. , 95 : 3 (1960) pp. 387–407
[3] J.A. Jenkins, "The general coefficient theorem and certain applications" Bull. Amer. Math. Soc. , 68 : 1 (1962) pp. 1–9
[4] J.A. Jenkins, "Some area theorems and a special coefficient theorem" Illinois J. Math. , 8 : 1 (1964) pp. 80–99
[5] P.M. Tamrazov, "On the general coefficient theorem" Math. USSR Sb. , 1 (1967) pp. 49–59 Mat. Sb. , 72 : 1 (1967) pp. 59–71
How to Cite This Entry:
Jenkins theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jenkins_theorem&oldid=13018
This article was adapted from an original article by P.M. Tamrazov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article