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 $:

$$ \left | \begin{array}{llllllll} 1 & 0 &\cdots & 0 &c _ { 0 } &c _ { 1 } &\cdots &c _ { {k-1} } \\ 0 & 1 &\cdots & 0 & 0 &c _ { 0 } &\cdots &c _ { {k-2} } \\ \vdots &\vdots &\ddots &\vdots &\vdots &\vdots &\ddots &\vdots \\ 0 & 0 &\cdots & 1 & 0 & 0 &\cdots &c _ { 0 } \\ \overline{c _ { 0 } } & 0 &\cdots & 0 & 1 & 0 &\cdots & 0 \\ \overline{c _ { 1 } } &\overline{c _ { 0 } } &\cdots & 0 & 0 & 1 &\cdots & 0 \\ \vdots &\vdots &\ddots &\vdots &\vdots &\vdots &\ddots &\vdots \\ \overline{c _ { {k-1} } } &\overline{c _ { {k-2} } } &\cdots &\overline{c _ { 0 } } & 0 & 0 &\cdots & 1 \\ \end{array} \right | > 0. $$

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