Jürgen Grahl

Hayman’s Alternative and Normal Families of Non-vanishing Meromorphic Functions

We show that if

Some applications of Cartan's theorem to normality and semiduality of gap power series

\cal F

A Modification of the Nevanlinna Theory

is a family of non-vanishing meromorphic functions in the unit disk D with P[f](z) ≠ 1 for all z ∈ D and all

Semiduality of Small Sets of Analytic Functions

f \in {\cal F}

Quasi-normality induced by differential inequalities

where P is a differential polynomial satisfying certain conditions, then

On the Growth of Real Functions and their Derivatives

\cal F

Differential polynomials which share a value with their derivative

; is normal. This generalizes former results of W. Schwick [19] and of M.-L. Fang [6]. Furthermore, we give the corresponding Picard type theorems generalizing Hayman’s Alternative.

Some new results on the semiduality of small sets of analytic functions

In this paper we give partial solutions to some questions concerning analytic functions with AP-Gaps raised by Pinto, Ruscheweyh and Salinas (cf. [9], [12]) using a theorem of H. Cartan which extends Montels theorem on analytic functions omitting the values 0 and 1. Using the same method, we also prove a generalization of a theorem in [9] on the dual hull of sets containing two elements.

Spherical derivatives and normal families

We present a modification of the Nevanlinna theory which is inspired by previous work of H. Cartan and D. Drasin and which makes full use of Poisson-Jensen-Nevanlinna’s Formula. We show that the results from the “classical” Nevanlinna theory remain valid for this modification. Furthermore, we give an estimate for the modified proximity function of the logarithmic derivatives of functions in a non-normal family which extends previous results by Drasin and W. Schwick.

Entire functions sharing a polynomial with their derivatives and normal families

We give some applications of normality criteria to the question of semiduality of sets of analytic functions consisting of two elements.