Generalization of Results about the Bohr Radius for Power Series

Generalization of Caratheodory’s Inequality and the Bohr Radius for Multidimensional Power Series

Remarks on the Bohr Phenomenon

Bohr’s theorem ([10]) states that analytic functions bounded by 1 in the unit disk have power series ∑ anz n such that ∑ |an||z| < 1 in the disk of radius 1/3 (the so-called Bohr radius.) On the other hand, it is known that there is no such Bohr phenomenon in Hardy spaces with the usual norm, although it is possible to build equivalent norms for which a Bohr phenomenon does occur! In this paper...

Bohr’s Power Series Theorem in Several Variables

Generalizing a classical one-variable theorem of Bohr, we show that if an n-variable power series has modulus less than 1 in the unit polydisc, then the sum of the moduli of the terms is less than 1 in the polydisc of radius 1/(3 √ n ). How large can the sum of the moduli of the terms of a convergent power series be? Harald Bohr addressed this question in 1914 with the following remarkable resu...

2 1 A pr 1 99 8 Multidimensional analogues of Bohr ’ s theorem on power series ∗

Generalizing the classical result of Bohr, we show that if an nvariable power series converges in an n-circular bounded complete domain D and its sum has modulus less than 1, then the sum of the maximum of the moduli of the terms is less than 1 in the homothetic domain r · D, where r = 1 − n √ 2/3. This constant is near to the best one for the domain D = {z : |z1| + . . . + |zn| < 1}. 1 Prelimi...

Bohr and Rogosinski Abscissas for Ordinary Dirichlet Series

We prove that the abscissas of Bohr and Rogosinski for ordinary Dirichlet series, mapping the right half-plane into the bounded convex domain G ⊂ C are independent of the domain G. Furthermore, we obtain new estimates about these abscissas. 1. Preliminaries Let us recall the theorem of H.Bohr [19] in 1914. Theorem 1.1. If a power series