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with at least one pole of order  $  \geq  2 $,  
 
with at least one pole of order  $  \geq  2 $,  
 
and let  $  P _ {1} \dots P _ {r} $
 
and let  $  P _ {1} \dots P _ {r} $
be all the poles of order 2, while  $  P _ {r+} 1 \dots P _ {p} $
+
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 $
 
are all the poles of orders higher than 2. Let an open and everywhere-dense set  $  \Delta $
 
on  $  {\mathcal R} $
 
on  $  {\mathcal R} $
Line 52: Line 52:
  
 
\begin{array}{ll}
 
\begin{array}{ll}
\alpha  ^ {(} j) z _ {j}  ^ {-} 2 +
+
\alpha  ^ {(j)} z _ {j}  ^ {-2} +
\textrm{ higher  powers  of  }  z _ {j}  ^ {-} 1 & \textrm{ if }  j \leq  r,  \\
+
\textrm{ higher  powers  of  }  z _ {j}  ^ {-1}  & \textrm{ if }  j \leq  r,  \\
\alpha  ^ {(} j) \left [ z _ {j} ^ {m _ {j} - 4 } + \sum _ {s =
+
\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 ;  \\
+
1 } ^  \infty  \beta _ {s}  ^ {(j)} z _ {j} ^ {m _ {j} - s - 4 } \right ]  & \textrm{ if }  j > r ;  \\
 
\end{array}
 
\end{array}
  
Line 63: Line 63:
 
f _ {0} ( z _ {j} )  =  \left \{  
 
f _ {0} ( z _ {j} )  =  \left \{  
 
\begin{array}{ll}
 
\begin{array}{ll}
\alpha  ^ {(} j) z _ {j} + \textrm{ non- positive  powers  of  }  z _ {j}  & \textrm{ if }  j \leq  r ,  \\
+
\alpha  ^ {(j)} z _ {j} + \textrm{ non- positive  powers  of  }  z _ {j}  & \textrm{ if }  j \leq  r ,  \\
 
z _ {j} +
 
z _ {j} +
\sum _ {s = n _ {j} } ^  \infty  a _ {s}  ^ {(} j) z _ {j}  ^ {-} s & \textrm{ if }  j> r ,  \\
+
\sum _ {s = n _ {j} } ^  \infty  a _ {s}  ^ {(j)} z _ {j}  ^ {-s}  & \textrm{ if }  j> r ,  \\
 
\end{array}
 
\end{array}
  
Line 86: Line 86:
  
 
$$  
 
$$  
T _ {j}  =  \alpha  ^ {(} j) \left [ a _ {m _ {j}  - 3 }  ^ {(} j) +
+
T _ {j}  =  \alpha  ^ {(j)} \left [ a _ {m _ {j}  - 3 }  ^ {(j)} +
\sum _ { s = 1 } ^ { {n _ j } } \beta _ {s}  ^ {(} j)
+
\sum _ { s = 1 } ^ { {n _ j } } \beta _ {s}  ^ {(j)}
a _ {m _ {j}  - s - 3 }  ^ {(} j) \right ] +
+
a _ {m _ {j}  - s - 3 }  ^ {(j)} \right ] +
 
$$
 
$$
  
Line 96: Line 96:
 
\begin{array}{ll}
 
\begin{array}{ll}
 
  0  & \textrm{ for  odd  }  m _ {j} ,  \\
 
  0  & \textrm{ for  odd  }  m _ {j} ,  \\
\alpha  ^ {(} j) \left (
+
\alpha  ^ {(j)} \left (
  
 
\frac{m _ {j} }{4}
 
\frac{m _ {j} }{4}
  - 1 \right ) \left ( a _ {n _ {j}  }  ^ {(} j)
+
  - 1 \right ) \left ( a _ {n _ {j}  }  ^ {(j)}
 
\right )  ^ {2}  & \textrm{ for  even  }  m _ {j} .  \\
 
\right )  ^ {2}  & \textrm{ for  even  }  m _ {j} .  \\
 
\end{array}
 
\end{array}
Line 108: Line 108:
  
 
$$ \tag{* }
 
$$ \tag{* }
  \mathop{\rm Re} \left \{ \sum _ {j = 1 } ^ { r }  \alpha  ^ {(} j)
+
  \mathop{\rm Re} \left \{ \sum _ {j = 1 } ^ { r }  \alpha  ^ {(j)}
  \mathop{\rm ln}  a  ^ {(} j) + \sum _ {j = r + 1 } ^ { p }  
+
  \mathop{\rm ln}  a  ^ {(j)} + \sum _ {j = r + 1 } ^ { p }  
\alpha  ^ {(} j) T _ {j} \right \}  \leq  0 ,
+
\alpha  ^ {(j)} T _ {j} \right \}  \leq  0 ,
 
$$
 
$$
  
where  $  \mathop{\rm ln}  a  ^ {(} j)  =  \mathop{\rm ln}  | a  ^ {(} j) | - i d ( P _ {j} ) $,  
+
where  $  \mathop{\rm ln}  a  ^ {(j)} =  \mathop{\rm ln}  | a  ^ {(j)} | - i d ( P _ {j} ) $,  
 
$  j \leq  r $.
 
$  j \leq  r $.
  
Line 132: Line 132:
 
$  j > r $,  
 
$  j > r $,  
 
of order  $  m _ {j} $
 
of order  $  m _ {j} $
such that  $  a _ {s}  ^ {(} j) = 0 $
+
such that  $  a _ {s}  ^ {(j)} = 0 $
 
for  $  s < \min ( n _ {j} + 1 , m _ {j} - 3 ) $;
 
for  $  s < \min ( n _ {j} + 1 , m _ {j} - 3 ) $;
  
Line 138: Line 138:
 
contains a pole  $  P _ {j} $
 
contains a pole  $  P _ {j} $
 
for which  $  j \leq  r $
 
for which  $  j \leq  r $
and  $  a  ^ {(} j) = 1 $;
+
and  $  a  ^ {(j)} = 1 $;
  
 
3)  $  \Delta _ {1} $
 
3)  $  \Delta _ {1} $
 
contains a point of a trajectory ending in a simple pole.
 
contains a point of a trajectory ending in a simple pole.
  
If (*) is an equality and if  $  | a  ^ {(} j) | \neq 1 $
+
If (*) is an equality and if  $  | a  ^ {(j)} | \neq 1 $
 
for a certain  $  j \leq  r $,  
 
for a certain  $  j \leq  r $,  
 
then  $  {\mathcal R} $
 
then  $  {\mathcal R} $
Line 152: Line 152:
 
is conformally equivalent to a linear mapping all fixed points of which are images of the poles.
 
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=55060
This article was adapted from an original article by P.M. Tamrazov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article