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Theorems for finding a solution to the [[Coefficient problem|coefficient problem]] for bounded analytic functions. They were obtained by I. Schur [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Schur,   "Ueber Potentzreihen, die im Innern des Einheitkreises berchränkt sind"  ''J. Reine Angew. Math.'' , '''147'''  (1917)  pp. 205–232</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Bieberbach,   "Lehrbuch der Funktionentheorie" , '''2''' , Teubner  (1931)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,   "Univalent functions" , Springer  (1983)  pp. Sect. 10.11</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.B. Garnett,   "Bounded analytic functions" , Acad. Press  (1981)  pp. 40</TD></TR></table>