Subordination principle
One of the forms of the Lindelöf principle, which employs the concept of subordination of functions. Let be any function regular in the disc
and satisfying the conditions
,
in
; let
be a meromorphic function in
. If the function
has the form
, then
is called subordinate to the function
in the disc
, while
is called the subordinating function. This subordination relation is denoted by
. In the simplest case where
is a univalent function in
, this relation simply means that
and that
does not take any values in the disc
that are not taken there by
. The following basic theorems apply.
Theorem 1.
Let the function be meromorphic in the disc
and map it on the Riemann surface
. Let
be the part of
corresponding to
,
. If
, then the values of
in
(under analytic continuation from
) lie in
, and the boundary points in
are obtained only for
,
[2].
Theorem 2.
If and if
is regular in
,
, then setting
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one has ,
,
, [1].
Theorem 3.
If and
is regular at
, then for the coefficients of the expansions
,
one has
,
[2].
The general theory of subordination is useful in considering the set of values taken or produced by an analytic function. The subordination relation can be used in two different ways. First, one can start from a given function and examine the behaviour of all
subordinate to
. If
is completely known, then the region
in which the values of
lie is also known (Theorem 1) as well as an upper bound on the integral means
(Theorem 2). If also
is regular at
, there are upper bounds for the coefficients of
(Theorem 3). Secondly, one can consider a function
that is meromorphic in the disc
and whose properties imply that it cannot be subordinate to a given function
in
. If here
, for example, is univalent, then
necessarily takes values outside
in
. These ideas of using the subordination relation illustrate the subordination principle and can be extended to multiply-connected domains [3].
References
[1] | J.E. Littlewood, "On inequalities in the theory of functions" Proc. Lond. Math. Soc. (2) , 23 (1925) pp. 481–519 |
[2] | W. Rogosinski, "On subordinate functions" Proc. Cambridge Philos. Soc. , 35 (1939) pp. 1–26 |
[3] | Yu. Alenitsyn, "A generalization of the subordination principle to multiply-connected domains" Trudy Mat. Inst. Steklov. , 60 (1961) pp. 5–21 (In Russian) |
[4] | W. Rogosinski, Schr. K. Gelehrt. Gesellsch. Naturwiss. Kl. , 8 : 1 (1931) pp. 1–31 |
[5] | W. Rogosinski, "On a theorem of Bieberbach–Eilenberg" J. Lond. Math. Soc. , 14 : 53 (1939) pp. 4–11 |
[6] | W. Rogosinski, "On the coefficients of subordinate functions" Proc. London Math. Soc. , 48 (1943) pp. 48–82 |
[7] | J.E. Littlewood, "Lectures on the theory of functions" , Oxford Univ. Press (1944) |
[8] | G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) |
Comments
References
[a1] | P.L. Duren, "Theory of ![]() |
Subordination principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subordination_principle&oldid=48897