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''on power series''
 
''on power series''
  
 
If a [[Power series|power series]]
 
If a [[Power series|power series]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b1203401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
\begin{equation} \tag{a1} \sum _ { k = 0 } ^ { \infty } c _ { k } z ^ { k } \end{equation}
  
converges in the unit disc and its sum has modulus less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b1203402.png" />, then
+
converges in the unit disc and its sum has modulus less than $1$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b1203403.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\begin{equation} \tag{a2} \sum _ { k = 0 } ^ { \infty } \left| c _ { k } z ^ { k } \right| &lt; 1 \end{equation}
  
in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b1203404.png" />. Moreover, the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b1203405.png" /> cannot be improved.
+
in the disc $\{ z : | z | &lt; 1 / 3 \}$. Moreover, the constant $1 / 3$ cannot be improved.
  
 
This formulation of the result of H. Bohr [[#References|[a1]]] is due to the work of M. Riesz, I. Schur and F. Wiener.
 
This formulation of the result of H. Bohr [[#References|[a1]]] is due to the work of M. Riesz, I. Schur and F. Wiener.
  
 
==Multi-dimensional variations.==
 
==Multi-dimensional variations.==
Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b1203406.png" /> the largest number such that if the series
+
Denote by $K_n$ the largest number such that if the series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b1203407.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
\begin{equation} \tag{a3} \sum _ { \alpha } c _ { \alpha } z ^ { \alpha } \end{equation}
  
converges in the unit poly-disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b1203408.png" /> and the estimate
+
converges in the unit poly-disc $U _ { 1 } = \{ z : | z _ { j } | &lt; 1 ,\; j = 1 , \ldots , n \}$ and the estimate
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b1203409.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
\begin{equation} \tag{a4} \left| \sum _ { \alpha } c _ { \alpha } z ^ { \alpha } \right| &lt; 1, \end{equation}
  
 
is valid there, then
 
is valid there, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
+
\begin{equation} \tag{a5} \sum _ { \alpha } | c _ { \alpha } z ^ { \alpha } | &lt; 1 \end{equation}
  
holds in the homothetic domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034011.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034012.png" />, all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034013.png" /> are non-negative integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034015.png" />.
+
holds in the homothetic domain $K _ { n } . U _ { 1 }$; here $\alpha = ( \alpha _ { 1 } , \ldots , \alpha _ { n } )$, all $\alpha_j$ are non-negative integers, $z = ( z_ 1 , \dots , z _ { n } )$, $z ^ { \alpha } = z _ { 1 } ^ { \alpha _ { 1 } } \ldots z _ { n } ^ { \alpha _ { n } }$.
  
Regarding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034016.png" />, the following is known [[#References|[a2]]]: For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034017.png" /> one has
+
Regarding $K_n$, the following is known [[#References|[a2]]]: For $n &gt; 1$ one has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
+
\begin{equation} \tag{a6} \frac { 1 } { 3 \sqrt { n } } &lt; K _ { n } &lt; \frac { 2 \sqrt { \operatorname { log } n } } { \sqrt { n } }. \end{equation}
  
Next, for the hypercone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034019.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034020.png" /> be the largest number such that if the series (a3) converges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034021.png" /> and the estimate (a4) is valid there, then (a5) holds in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034022.png" />.
+
Next, for the hypercone $D ^ { 0 } = \{ z : | z _ { 1 } | + \ldots + | z _ { n } | &lt; 1 \}$, let $K _ { n } ( D ^ { \circ } )$ be the largest number such that if the series (a3) converges in $D ^ { \circ }$ and the estimate (a4) is valid there, then (a5) holds in $K _ { n } ( D ^ { \circ } ) . D ^ { \circ }$.
  
For the hypercone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034023.png" /> the following estimates are true [[#References|[a3]]]:
+
For the hypercone $D ^ { \circ }$ the following estimates are true [[#References|[a3]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034024.png" /></td> </tr></table>
+
\begin{equation*} \frac { 1 } { 3 e ^ { 1 / 3 } } &lt; K _ { n } ( D ^ { \circ } ) \leq \frac { 1 } { 3 }. \end{equation*}
  
Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034025.png" />, then there exists a series of the form (a3) converging in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034026.png" /> and such that the estimate (a4) is valid there, but (a5) fails at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034027.png" />.
+
Moreover, if $z \notin 1 / 3 \cdot D ^ { \circ }$, then there exists a series of the form (a3) converging in $D ^ { \circ }$ and such that the estimate (a4) is valid there, but (a5) fails at the point $z$.
  
Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034028.png" /> the largest number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034029.png" /> such that if the series (a3) converges in a complete Reinhardt bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034030.png" /> and (a4) holds in it, then
+
Denote by $B _ { n } ( D )$ the largest number $r$ such that if the series (a3) converges in a complete Reinhardt bounded domain $D$ and (a4) holds in it, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
+
\begin{equation} \tag{a7} \sum _ { \alpha } \operatorname { sup } _ { D _ { r } } | c _ { \alpha } z ^ { \alpha } | &lt; 1, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034032.png" /> is a homothetic transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034033.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034034.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034035.png" />. This gives a natural generalization of Bohr's theorem.
+
where $D _ { r } = . D$ is a homothetic transform of $D$. If $D = U _ { 1 }$, then $B _ { n } ( D ) = K _ { n }$. This gives a natural generalization of Bohr's theorem.
  
 
The inequality
 
The inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034036.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>
+
\begin{equation} \tag{a8} 1 - \sqrt [ \frac { 2 } { 3 } ] { n } &lt; B _ { n } ( D ). \end{equation}
  
is true for any complete bounded Reinhardt domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034037.png" /> [[#References|[a3]]].
+
is true for any complete bounded Reinhardt domain $D$ [[#References|[a3]]].
  
This estimate can be improved for concrete domains [[#References|[a3]]]: For the unit hypercone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034038.png" /> the following inequality holds:
+
This estimate can be improved for concrete domains [[#References|[a3]]]: For the unit hypercone $D ^ { \circ }$ the following inequality holds:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>
+
\begin{equation} \tag{a9} B _ { n } ( D ^ { \circ } ) &lt; \frac { 0.446663 } { n }. \end{equation}
  
 
==Arbitrary bases.==
 
==Arbitrary bases.==
In [[#References|[a4]]], Bohr's phenomenon was studied for arbitrary bases in the space of holomorphic functions on an arbitrary domain, by analogy with (a7) (or (a5)). One can easily see that Bohr's phenomenon appears for a given basis only if the basis contains a constant function. It has been proven that if, in addition, all other functions of the basis vanish at some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034040.png" />, then there exist a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034041.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034042.png" /> and a compact subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034043.png" /> such that, whenever a holomorphic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034044.png" /> has modulus less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034045.png" />, the sum of the maximum in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034046.png" /> of the moduli of the terms of its expansion is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034047.png" /> too.
+
In [[#References|[a4]]], Bohr's phenomenon was studied for arbitrary bases in the space of holomorphic functions on an arbitrary domain, by analogy with (a7) (or (a5)). One can easily see that Bohr's phenomenon appears for a given basis only if the basis contains a constant function. It has been proven that if, in addition, all other functions of the basis vanish at some point $z _ { 0 } \in D$, then there exist a neighbourhood $U$ of $z_0$ and a compact subset $K \subset D$ such that, whenever a holomorphic function on $K$ has modulus less than $1$, the sum of the maximum in $U$ of the moduli of the terms of its expansion is less than $1$ too.
  
More precisely, one has proven [[#References|[a4]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034048.png" /> is a [[Complex manifold|complex manifold]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034049.png" /> is a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034050.png" /> satisfying:
+
More precisely, one has proven [[#References|[a4]]] that if $M$ is a [[Complex manifold|complex manifold]] and $( \varphi _ { n } ) _ { n = 0 } ^ { \infty }$ is a basis in $H ( M )$ satisfying:
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034051.png" />;
+
i) $\varphi _ { 0 } = 1$;
  
ii) there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034052.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034054.png" />, then there exist a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034055.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034056.png" /> and a compact subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034057.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034059.png" />,
+
ii) there exists a $z _ { 0 } \in M$ such that $\varphi _ { n } ( z _ { 0 } ) = 0$, $n = 1,2 , \dots$, then there exist a neighbourhood $U$ of $z_0$ and a compact subset $K \subset M$ such that for all $f \in H ( M )$, $f = \sum f _ { n } \varphi _ { n }$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034060.png" /></td> </tr></table>
+
\begin{equation*} \sum _ { 0 } ^ { \infty } | f _ { n } | \operatorname { sup } _ { U } | \varphi _ { n } ( z ) | \leq \operatorname { sup } _ { K } | f ( z ) |. \end{equation*}
  
 
For holomorphic functions with positive real part the following assertion (analogous to the initial formulation) holds [[#References|[a5]]]. If the function
 
For holomorphic functions with positive real part the following assertion (analogous to the initial formulation) holds [[#References|[a5]]]. If the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034061.png" /></td> </tr></table>
+
\begin{equation*} f ( z ) = \sum _ { k = 0 } ^ { \infty } c _ { k } z ^ { k } , \quad | z | &lt; 1, \end{equation*}
  
has positive real part and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034062.png" />, then
+
has positive real part and $f ( 0 ) &gt; 0$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034063.png" /></td> </tr></table>
+
\begin{equation*} \sum _ { k = 0 } ^ { \infty } | c  _ { k } z ^ { k } | &lt; 2 f ( 0 ) \end{equation*}
  
in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034064.png" /> and the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034065.png" /> cannot be improved.
+
in the disc $\{ z : | z | &lt; 1 / 3 \}$ and the constant $1 / 3$ cannot be improved.
  
Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034066.png" /> is the unit disc and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034067.png" />, the Bohr radius in the above assertion and that in the initial assertion are equal. The next results shows that Bohr radii are equal in a more general situation too [[#References|[a5]]].
+
Thus, if $M$ is the unit disc and $z _ { 0 } = 0$, the Bohr radius in the above assertion and that in the initial assertion are equal. The next results shows that Bohr radii are equal in a more general situation too [[#References|[a5]]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034068.png" /> be a complex manifold, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034069.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034070.png" /> be a continuous [[Semi-norm|semi-norm]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034071.png" /> such that
+
Let $M$ be a complex manifold, $z _ { 0 } \in M$ and let $|.|$ be a continuous [[Semi-norm|semi-norm]] in $H ( M )$ such that
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034072.png" />;
+
a) $\| f \| = | f ( z _ { 0 } ) | + \| f - f ( z _ { 0 } ) \|$;
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034073.png" />. Then the following statements are equivalent:
+
b) $\| f \cdot g \| \leq \| f \| \cdot \| g \|$. Then the following statements are equivalent:
  
A) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034074.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034075.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034077.png" />;
+
A) $\| f \| \leq 2 f ( z _ { 0 } )$ if $\operatorname { Re } f ( z ) &gt; 0$ for all $z \in M$ and $f ( z _ { 0 } ) &gt; 0$;
  
B) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034078.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120340/b12034079.png" />.
+
B) $\|\, f \| \leq \operatorname { sup } _ { M } |\, f ( z ) |$ for all $f \in H ( M )$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Bohr,  "A theorem concerning power series"  ''Proc. London Math. Soc.'' , '''13''' :  2  (1914)  pp. 1–5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.P. Boas,  D. Khavinson,  "Bohr's power series theorem in several variables"  ''Proc. Amer. Math. Soc.'' , '''125'''  (1997)  pp. 2975–2979</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Aizenberg,  "Multidimensional analogues of Bohr's theorem on power series"  ''Proc. Amer. Math. Soc.'' , '''128'''  (2000)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Aizenberg,  A. Aytuna,  P. Djakov,  "Generalization of Bohr's theorem for arbitrary bases in spaces of holomorphic functions of several variables"  ''J. Anal. Appl.''  (to appear)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L. Aizenberg,  A. Aytuna,  P. Djakov,  "An abstract approach to Bohr phenomenon"  ''Proc. Amer. Math. Soc.''  (to appear)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  H. Bohr,  "A theorem concerning power series"  ''Proc. London Math. Soc.'' , '''13''' :  2  (1914)  pp. 1–5</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  H.P. Boas,  D. Khavinson,  "Bohr's power series theorem in several variables"  ''Proc. Amer. Math. Soc.'' , '''125'''  (1997)  pp. 2975–2979</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  L. Aizenberg,  "Multidimensional analogues of Bohr's theorem on power series"  ''Proc. Amer. Math. Soc.'' , '''128'''  (2000)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  L. Aizenberg,  A. Aytuna,  P. Djakov,  "Generalization of Bohr's theorem for arbitrary bases in spaces of holomorphic functions of several variables"  ''J. Anal. Appl.''  (to appear)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  L. Aizenberg,  A. Aytuna,  P. Djakov,  "An abstract approach to Bohr phenomenon"  ''Proc. Amer. Math. Soc.''  (to appear)</td></tr></table>

Revision as of 17:03, 1 July 2020

on power series

If a power series

\begin{equation} \tag{a1} \sum _ { k = 0 } ^ { \infty } c _ { k } z ^ { k } \end{equation}

converges in the unit disc and its sum has modulus less than $1$, then

\begin{equation} \tag{a2} \sum _ { k = 0 } ^ { \infty } \left| c _ { k } z ^ { k } \right| < 1 \end{equation}

in the disc $\{ z : | z | < 1 / 3 \}$. Moreover, the constant $1 / 3$ cannot be improved.

This formulation of the result of H. Bohr [a1] is due to the work of M. Riesz, I. Schur and F. Wiener.

Multi-dimensional variations.

Denote by $K_n$ the largest number such that if the series

\begin{equation} \tag{a3} \sum _ { \alpha } c _ { \alpha } z ^ { \alpha } \end{equation}

converges in the unit poly-disc $U _ { 1 } = \{ z : | z _ { j } | < 1 ,\; j = 1 , \ldots , n \}$ and the estimate

\begin{equation} \tag{a4} \left| \sum _ { \alpha } c _ { \alpha } z ^ { \alpha } \right| < 1, \end{equation}

is valid there, then

\begin{equation} \tag{a5} \sum _ { \alpha } | c _ { \alpha } z ^ { \alpha } | < 1 \end{equation}

holds in the homothetic domain $K _ { n } . U _ { 1 }$; here $\alpha = ( \alpha _ { 1 } , \ldots , \alpha _ { n } )$, all $\alpha_j$ are non-negative integers, $z = ( z_ 1 , \dots , z _ { n } )$, $z ^ { \alpha } = z _ { 1 } ^ { \alpha _ { 1 } } \ldots z _ { n } ^ { \alpha _ { n } }$.

Regarding $K_n$, the following is known [a2]: For $n > 1$ one has

\begin{equation} \tag{a6} \frac { 1 } { 3 \sqrt { n } } < K _ { n } < \frac { 2 \sqrt { \operatorname { log } n } } { \sqrt { n } }. \end{equation}

Next, for the hypercone $D ^ { 0 } = \{ z : | z _ { 1 } | + \ldots + | z _ { n } | < 1 \}$, let $K _ { n } ( D ^ { \circ } )$ be the largest number such that if the series (a3) converges in $D ^ { \circ }$ and the estimate (a4) is valid there, then (a5) holds in $K _ { n } ( D ^ { \circ } ) . D ^ { \circ }$.

For the hypercone $D ^ { \circ }$ the following estimates are true [a3]:

\begin{equation*} \frac { 1 } { 3 e ^ { 1 / 3 } } < K _ { n } ( D ^ { \circ } ) \leq \frac { 1 } { 3 }. \end{equation*}

Moreover, if $z \notin 1 / 3 \cdot D ^ { \circ }$, then there exists a series of the form (a3) converging in $D ^ { \circ }$ and such that the estimate (a4) is valid there, but (a5) fails at the point $z$.

Denote by $B _ { n } ( D )$ the largest number $r$ such that if the series (a3) converges in a complete Reinhardt bounded domain $D$ and (a4) holds in it, then

\begin{equation} \tag{a7} \sum _ { \alpha } \operatorname { sup } _ { D _ { r } } | c _ { \alpha } z ^ { \alpha } | < 1, \end{equation}

where $D _ { r } = r . D$ is a homothetic transform of $D$. If $D = U _ { 1 }$, then $B _ { n } ( D ) = K _ { n }$. This gives a natural generalization of Bohr's theorem.

The inequality

\begin{equation} \tag{a8} 1 - \sqrt [ \frac { 2 } { 3 } ] { n } < B _ { n } ( D ). \end{equation}

is true for any complete bounded Reinhardt domain $D$ [a3].

This estimate can be improved for concrete domains [a3]: For the unit hypercone $D ^ { \circ }$ the following inequality holds:

\begin{equation} \tag{a9} B _ { n } ( D ^ { \circ } ) < \frac { 0.446663 } { n }. \end{equation}

Arbitrary bases.

In [a4], Bohr's phenomenon was studied for arbitrary bases in the space of holomorphic functions on an arbitrary domain, by analogy with (a7) (or (a5)). One can easily see that Bohr's phenomenon appears for a given basis only if the basis contains a constant function. It has been proven that if, in addition, all other functions of the basis vanish at some point $z _ { 0 } \in D$, then there exist a neighbourhood $U$ of $z_0$ and a compact subset $K \subset D$ such that, whenever a holomorphic function on $K$ has modulus less than $1$, the sum of the maximum in $U$ of the moduli of the terms of its expansion is less than $1$ too.

More precisely, one has proven [a4] that if $M$ is a complex manifold and $( \varphi _ { n } ) _ { n = 0 } ^ { \infty }$ is a basis in $H ( M )$ satisfying:

i) $\varphi _ { 0 } = 1$;

ii) there exists a $z _ { 0 } \in M$ such that $\varphi _ { n } ( z _ { 0 } ) = 0$, $n = 1,2 , \dots$, then there exist a neighbourhood $U$ of $z_0$ and a compact subset $K \subset M$ such that for all $f \in H ( M )$, $f = \sum f _ { n } \varphi _ { n }$,

\begin{equation*} \sum _ { 0 } ^ { \infty } | f _ { n } | \operatorname { sup } _ { U } | \varphi _ { n } ( z ) | \leq \operatorname { sup } _ { K } | f ( z ) |. \end{equation*}

For holomorphic functions with positive real part the following assertion (analogous to the initial formulation) holds [a5]. If the function

\begin{equation*} f ( z ) = \sum _ { k = 0 } ^ { \infty } c _ { k } z ^ { k } , \quad | z | < 1, \end{equation*}

has positive real part and $f ( 0 ) > 0$, then

\begin{equation*} \sum _ { k = 0 } ^ { \infty } | c _ { k } z ^ { k } | < 2 f ( 0 ) \end{equation*}

in the disc $\{ z : | z | < 1 / 3 \}$ and the constant $1 / 3$ cannot be improved.

Thus, if $M$ is the unit disc and $z _ { 0 } = 0$, the Bohr radius in the above assertion and that in the initial assertion are equal. The next results shows that Bohr radii are equal in a more general situation too [a5].

Let $M$ be a complex manifold, $z _ { 0 } \in M$ and let $|.|$ be a continuous semi-norm in $H ( M )$ such that

a) $\| f \| = | f ( z _ { 0 } ) | + \| f - f ( z _ { 0 } ) \|$;

b) $\| f \cdot g \| \leq \| f \| \cdot \| g \|$. Then the following statements are equivalent:

A) $\| f \| \leq 2 f ( z _ { 0 } )$ if $\operatorname { Re } f ( z ) > 0$ for all $z \in M$ and $f ( z _ { 0 } ) > 0$;

B) $\|\, f \| \leq \operatorname { sup } _ { M } |\, f ( z ) |$ for all $f \in H ( M )$.

References

[a1] H. Bohr, "A theorem concerning power series" Proc. London Math. Soc. , 13 : 2 (1914) pp. 1–5
[a2] H.P. Boas, D. Khavinson, "Bohr's power series theorem in several variables" Proc. Amer. Math. Soc. , 125 (1997) pp. 2975–2979
[a3] L. Aizenberg, "Multidimensional analogues of Bohr's theorem on power series" Proc. Amer. Math. Soc. , 128 (2000)
[a4] L. Aizenberg, A. Aytuna, P. Djakov, "Generalization of Bohr's theorem for arbitrary bases in spaces of holomorphic functions of several variables" J. Anal. Appl. (to appear)
[a5] L. Aizenberg, A. Aytuna, P. Djakov, "An abstract approach to Bohr phenomenon" Proc. Amer. Math. Soc. (to appear)
How to Cite This Entry:
Bohr theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bohr_theorem&oldid=12334
This article was adapted from an original article by L. Aizenberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article