Bohr theorem

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)