Dmitry Khavinson

Bohr's power series theorem in several variables

Generalizing a classical one-variable theorem of Harald 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^{1/2}).

On the number of zeros of certain rational harmonic functions

Extending a result of Khavinson and Swiatek (2003) we show that the rational harmonic function r(z) - z, where r(z) is a rational function of degree n > 1, has no more than 5n - 5 complex zeros. Applications to gravitational lensing are discussed. In particular, this result settles a conjecture by Rhie concerning the maximum number of lensed images due to an n-point gravitational lens.

Remarks on the Bohr Phenomenon

Bohr’s Theorem [10] states that analytic functions bounded by 1 in the unit disk have power series

On the number of zeros of certain harmonic polynomials

\sum a_n z^n

Gravitational Lensing by Elliptical Galaxies, and the Schwarz Function

such that

On the classical Dirichlet problem in the plane with rational data

\sum |a_n||z|^n<1

Recurrence Relations for Orthogonal Polynomials and Algebraicity of Solutions of the Dirichlet Problem

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, we consider Hardy space functions that vanish at the origin and obtain an exact positive Bohr radius. Also, following [4, 11], we discuss the growth and Bohr phenomena for series of the type

ON ANNIHILATORS OF HARMONIC VECTOR FIELDS

\sum |a_n|^p r^n

On point to point reflection of harmonic functions across real-analytic hypersurfaces in ℝ n

,

Annihilating measures of the algebra R(X)

0<p<2