Schur theorems
Theorems for finding a solution to the coefficient problem for bounded analytic functions. They were obtained by I. Schur [1]. Let be the class of regular functions
in
satisfying in it the condition
. Let
,
, be the
-dimensional complex Euclidean space, its points are
-tuples of complex numbers
; let
be a set of points
such that the numbers
are the first
coefficients of some function from
. The sets
are closed, bounded and convex in
. Then the following theorems hold.
Schur's first theorem: To the points on the boundary of
there correspond in
only rational functions of the form
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Schur's second theorem: A necessary and sufficient condition for to be an interior point of
is that the following inequalities hold for
:
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Schur's second theorem provides the complete solution to the coefficient problem for bounded functions in the case of interior points of the coefficients region.
References
[1] | I. Schur, "Ueber Potentzreihen, die im Innern des Einheitkreises berchränkt sind" J. Reine Angew. Math. , 147 (1917) pp. 205–232 |
[2] | L. Bieberbach, "Lehrbuch der Funktionentheorie" , 2 , Teubner (1931) |
[3] | 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, "Univalent functions" , Springer (1983) pp. Sect. 10.11 |
[a2] | J.B. Garnett, "Bounded analytic functions" , Acad. Press (1981) pp. 40 |
Schur theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schur_theorems&oldid=12059