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 be a finite oriented Riemann surface, let
be a positive quadratic differential on
with at least one pole of order
, and let
be all the poles of order 2, while
are all the poles of orders higher than 2. Let an open and everywhere-dense set
on
be the complement of the union of a finite number of closures of trajectories and closures of trajectory arcs, and let
,
. Suppose the function
maps
conformally and univalently (cf. Conformal mapping) into
, and suppose there exists a homotopy
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of the mapping into the identity mapping
which leaves all poles from
fixed and satisfies the condition
for each pole
,
, and each point
. Let
be a local parameter for the pole
such that
,
. Let, for
, in a neighbourhood of the pole
,
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where is the integer part of the number
. Let
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and let for all
for which
lies on the boundary of a strip-like domain with respect to
. Finally, let
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The statement of Jenkins' theorem. Under the conditions mentioned above,
![]() | (*) |
where ,
.
Jenkins' theorem in the case of equality. If in (*) the equality sign holds, then: a) in each domain the mapping
is an isometry in the
-metric:
, each trajectory
in
is mapped to a trajectory, and the set
is everywhere-dense in
; b) for
to be the identity mapping in a certain domain
it is sufficient that one of the following additional conditions holds:
1) contains a pole
,
, of order
such that
for
;
2) contains a pole
for which
and
;
3) contains a point of a trajectory ending in a simple pole.
If (*) is an equality and if for a certain
, then
is conformally equivalent to the sphere,
has no zeros or simple poles and
. If, in addition,
is a domain, the mapping
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 |
Jenkins theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jenkins_theorem&oldid=47464