Jenkins 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, [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