Schur theorems

Theorems for finding a solution to the coefficient problem for bounded analytic functions. They were obtained by I. Schur [1]. Let $ B $ be the class of regular functions $ f( z) = c _ {0} + c _ {1} z + \dots $ in $ | z | < 1 $ satisfying in it the condition $ | f( z ) | \leq 1 $. Let $ \mathbf C ^ {n} $, $ n \geq 1 $, be the $ n $- dimensional complex Euclidean space, its points are $ n $- tuples of complex numbers $ ( c _ {0} \dots c _ {n-} 1 ) $; let $ B ^ {(} n) $ be a set of points $ ( c _ {0} \dots c _ {n-} 1 ) \in \mathbf C ^ {n} $ such that the numbers $ c _ {0} \dots c _ {n-} 1 $ are the first $ n $ coefficients of some function from $ B $. The sets $ B ^ {(} n) $ are closed, bounded and convex in $ \mathbf C ^ {n} $. Then the following theorems hold.

Schur's first theorem: To the points $ ( c _ {0} \dots c _ {n-} 1 ) $ on the boundary of $ B ^ {(} n) $ there correspond in $ B $ only rational functions of the form

$$ \frac{\overline{ {\alpha _ {n-} 1 }}\; + \overline{ {\alpha _ {n-} 2 }}\; z + \dots + \overline{ {\alpha _ {0} }}\; z ^ {n-} 1 }{\alpha _ {0} + \alpha _ {1} z + \dots + \alpha _ {n-} 1 z ^ {n-} 1 } . $$

Schur's second theorem: A necessary and sufficient condition for $ ( c _ {0} \dots c _ {n-} 1 ) $ to be an interior point of $ B ^ {(} n) $ is that the following inequalities hold for $ k = 1 \dots n $:

$$ \left | \begin{array}{llllllll} 1 & 0 &\cdot & 0 &c _ {0} &c _ {1} &\cdot &c _ {k-} 1 \\ 0 & 1 &\cdot & 0 & 0 &c _ {0} &\cdot &c _ {k-} 2 \\ \cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot \\ 0 & 0 &\cdot & 1 & 0 & 0 &\cdot &c _ {0} \\ \overline{ {c _ {0} }}\; & 0 &\cdot & 0 & 1 & 0 &\cdot & 0 \\ \overline{ {c _ {1} }}\; &\overline{ {c _ {0} }}\; &\cdot & 0 & 0 & 1 &\cdot & 0 \\ \cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot \\ \overline{ {c _ {k-} 1 }}\; &\overline{ {c _ {k-} 2 }}\; &\cdot &{c _ {0} } bar & 0 & 0 &\cdot & 1 \\ \end{array} \right | > 0. $$

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)

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