Volume 28, Issue 4

A prime number p is called b -elite if only finitely many generalized Fermat numbers F b,n = b 2 n +1 are quadratic residues modulo p . We generalize a Theorem of Křížek, Luca, and Somer giving an asymptotic bound for elite primes, present some further results and derive conjectures concerning primes related to generalized elites.

The purpose of this article is to show a uniqueness theorem for meromorphic mappings of C m into C P n with moving targets and without counting multiplicities.

We study surfaces of prescribed mean curvature in R 3 with part of their boundaries lying on a support manifold without boundary. We prove C 1, μ -regularity of such a surface, whenever the support manifold is of class C 2 and the surface itself is a continuous, stationary point of the associated energy functional; consequently, minimizers of that functional are included. In addition, asymptotic expansions near boundary branch points are provided. Our results improve previous work of Hildebrandt and Jäger [HJ] and Hardt [Ha], and generalize corresponding theorems on minimal surfaces. The main difficulty arises from the fact that stationary surfaces with prescribed mean curvature do not have to meet the support manifold perpendicularly, in contrast to minimal surfaces which are stationary points of Dirichlet´s functional.

In this note we prove that the best constant in the Schwarz inequality for the class of squares of harmonic conjugate functions in the unit disc vanishing at the center is one.

Tests for the integrability of the Fourier transform of a function are given in terms of belonging of the function simultaneously to two spaces of smooth functions. These are, in a sense, generalizations of Zygmund´s test for the absolute convergence of Fourier series.

We show that Hayman´s theorem on the value distribution of meromorphic functions and their derivatives cannot be translated in a one-to-one fashion to the terms which are used in Ahlfors´ theory of covering surfaces. This answers a question in [2].

We discuss some known results and open problems pertaining to the nonexistence of nontrivial solutions for semilinear polyharmonic equations under Dirichlet or Navier boundary conditions. Precise control of the boundary terms in a Pohozaev-type identity allows us to establish a sharp nonexistence criterion for radially symmetric solutions, which closes a gap in the literature.