Difference between revisions of "Schur theorems"
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Let $ \mathbf C ^ { n } $, | Let $ \mathbf C ^ { n } $, | ||
$ n \geq 1 $, | $ n \geq 1 $, | ||
| − | be the $ n $- | + | be the $ n $-dimensional complex Euclidean space, its points are $ n $- |
| − | dimensional complex Euclidean space, its points are $ n $- | + | tuples of complex numbers $ { ( c _ { 0 }, \dots, c _ { {n-1} } ) } $; |
| − | tuples of complex numbers $ { ( c _ { 0 } \dots c _ { {n-1} } ) } $; | ||
let $ B ^ { { ( n) } } $ | let $ B ^ { { ( n) } } $ | ||
| − | be a set of points $ { ( c _ { 0 } | + | be a set of points $ { ( c _ { 0 }, \dots, c _ { {n-1} } ) } \in \mathbf C ^ { n } $ |
| − | such that the numbers $ c _ { 0 } | + | such that the numbers $ c _ { 0 }, \dots, c _ { {n-1} } $ |
are the first $ n $ | are the first $ n $ | ||
coefficients of some function from $ B $. | coefficients of some function from $ B $. | ||
| Line 29: | Line 28: | ||
Then the following theorems hold. | Then the following theorems hold. | ||
| − | Schur's first theorem: To the points $ { ( c _ { 0 } \dots c _ { {n-1} } ) } $ | + | Schur's first theorem: To the points $ { ( c _ { 0 }, \dots, c _ { {n-1} } ) } $ |
on the boundary of $ B ^ { { ( n) } } $ | on the boundary of $ B ^ { { ( n) } } $ | ||
there correspond in $ B $ | there correspond in $ B $ | ||
| Line 36: | Line 35: | ||
$$ \frac{ \overline{\alpha _ { {n-1} } } + \overline{\alpha _ { {n-2} } } {z+\dots} + \overline{\alpha _ { 0 } } z ^ { {n-1} } }{ \alpha _ { 0 } {+\alpha} _ { 1 } {z+\dots} {+\alpha} _ { {n-1} } z ^ { {n-1} } } . $$ | $$ \frac{ \overline{\alpha _ { {n-1} } } + \overline{\alpha _ { {n-2} } } {z+\dots} + \overline{\alpha _ { 0 } } z ^ { {n-1} } }{ \alpha _ { 0 } {+\alpha} _ { 1 } {z+\dots} {+\alpha} _ { {n-1} } z ^ { {n-1} } } . $$ | ||
| − | Schur's second theorem: A necessary and sufficient condition for $ { ( c _ { 0 } \dots c _ { {n-1} } ) } $ | + | Schur's second theorem: A necessary and sufficient condition for $ { ( c _ { 0 }, \dots, c _ { {n-1} } ) } $ |
to be an interior point of $ B ^ { { ( n) } } $ | to be an interior point of $ B ^ { { ( n) } } $ | ||
| − | is that the following inequalities hold for $ k = 1 \dots n $: | + | is that the following inequalities hold for $ k = 1, \dots, n $: |
| − | $$ \left | \begin{array}{llllllll} 1 & 0 &\ | + | $$ \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. $$ |
Schur's second theorem provides the complete solution to the coefficient problem for bounded functions in the case of interior points of the coefficients region. | Schur's second theorem provides the complete solution to the coefficient problem for bounded functions in the case of interior points of the coefficients region. | ||
Latest revision as of 00:51, 21 January 2022
Theorems for finding a solution to the coefficient problem for bounded analytic functions. They were obtained by I. Schur [1]. Let $ B $
be the class of regular functions $ f{ ( z) } = c _ { 0 } {+c} _ { 1 } {z+\dots} $
in $ | z | < 1 $
satisfying in it the condition $ | f{ ( z ) } | \leq 1 $.
Let $ \mathbf C ^ { n } $,
$ n \geq 1 $,
be the $ n $-dimensional complex Euclidean space, its points are $ n $-
tuples of complex numbers $ { ( c _ { 0 }, \dots, c _ { {n-1} } ) } $;
let $ B ^ { { ( n) } } $
be a set of points $ { ( c _ { 0 }, \dots, c _ { {n-1} } ) } \in \mathbf C ^ { n } $
such that the numbers $ c _ { 0 }, \dots, c _ { {n-1} } $
are the first $ n $
coefficients of some function from $ B $.
The sets $ B ^ { { ( n) } } $
are closed, bounded and convex in $ \mathbf C ^ { n } $.
Then the following theorems hold.
Schur's first theorem: To the points $ { ( c _ { 0 }, \dots, c _ { {n-1} } ) } $ on the boundary of $ B ^ { { ( n) } } $ there correspond in $ B $ only rational functions of the form
$$ \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) |
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=49678