Jörg Winkler

Über Picardmengen ganzer Funktionen

Following Lehto we call a set E in the complex plane a Picard set for integral functions, if every non-rational integral function omits at most one finite value in the complement (with respect to the plane) of E. The existence of non-trivial Picard sets was proved by Lehto [3]. The aim of this paper is to give a new criterion for denumberable point sets E to be Picard sets for integral functions. In some way the criterion given by theorem 1 is an extension of the result on Picard sets for integral functions given by Toppila [5] and improves the criterion given by the author in [6].

Some lower bounds for the maximal growth of the spherical derivative

This paper gives estimates for the growth of the spherical derivative of meromorphic functions compared with their Nevanlinna characteristic. Here the estimates are depending from the order respectively the growth of the functions itself.

Über den Verzweigungsindex bei ganzen Funktionen

Recently the author proved ([3]) for entire functions f(z) and any complex a: Each set of n a-points of f(z), which are near (with respect to the order of f(z)) to each another can be considered in view of the second fundamental theorem of Nevanlinna as an a-point of at least multiplicity two if n>1. In the present paper it will be proved: Each set of n a-points of f(z), which are very near (with respect to the order of f(z)) to each another can be considered in view of the second fundamental theorem of Nevanlinna as an a-point of at least multiplicity n−1.

On functions with strongly growing spherical derivatives

A strongly growing spherical derivative of meromorphic functions gives information about the local value distribution of that funclion. Here it is shown that a strongly growing spherical derivative is hercdrrablc to it derivative. unless thc function takes in certain sequence of discs all values at most once. but in any subsequence of these dises all values of the extended complex plane with one exception. To prove this result some interesting between the behavior of the function itself and its spherical drivative are considered.

Some remarks on some theorem of weierstraß

The theorem of Weierstras gives that any function meromorphic in the complex plane comes arbitrarily near to any point w e in each neighborhood of infinity. In this paper it will be proved for each given order ρ (of functions meromorphic in ) that there are sequences of points a 1 a 2… depending on ρ with the following property: Each function f transcendental and meromorphic in of order λ⩽ ρ comes arbitrarily near to any point we in the intersection of each neighborhood of infinity with {a 1 a 2…} if f has at least one deficiency in the sense of Nevanlinna. It will be shown that the suppositions are essentially sharp.