Gary G. Gundersen

Meromorphic functions that share four values

An old theorem of R. Nevanlinna states that if two distinct nonconstant meromorphic functions share four values counting multiplicities, then the functions are Mbbius transformations of each other, two of the shared values are Picard values for both functions, and the cross ratio of a particular permutation of the shared values equals -1. In this paper we show that if two nonconstant meromorphic functions share two values counting multiplicities and share two other values ignoring multiplicities, then the functions share all four values counting multiplicities.

The possible orders of solutions of linear differential equations with polynomial coefficients

We find specific inforrnation about the possible orders of transeendental solutions of equations of the form f(n) +Pn _l(z)f(n-l) ++Po(z)f0, where Po(Z),Pl(Z)? aPnL(Z) are polynomials with po(z) g 0. Several examples are given.

The Strength of Cartan's Version of Nevanlinna Theory

In 1933 Henri Cartan proved a fundamental theorem in Nevanlinna theory, namely a generalization of Nevanlinna’s second fundamental theorem. Cartan’s theorem works very well for certain kinds of problems. Unfortunately, it seems that Cartan’s theorem, its proof, and its usefulness, are not as widely known as they deserve to be. To help give wider exposure to Cartan’s theorem, the simple and general forms of the theorem are stated here. A proof of the general form is given, as well as several applications of the theorem.

Meromorphic functions that share three values IM and a fourth value CM

The main open question in the theory of meromorphic functions that share four values is: If two nonconstant meromorphic functions share three values ignoring multiplicities and share a fourth value counting multiplicities, then do the functions necessarily share all four values counting multiplicities? We prove a partial result on this question.

Subnormal Solutions of Second Order Linear Differential Equations With Periodic Coefficients

We find the form of all subnormal solutions of equation (1.4). Our results generalize and improve a well-known result of Wittich about equation (1.1). Several examples are given. Higher order equations are discussed.

Meromorphic functions that share pairs of values

It is well known that for meromorphic functions that share values (ignoring multiplicities), the “dividing line” between the trivial case and the nontrivial examples occurs between four and five shared values. In this paper we show that for meromorphic functions that share pairs of values (ignoring multiplicities), the “dividing line” between the trivial case and the nontrivial examples occurs between five and six pairs of values. We also discuss meromorphic functions that share four pairs of values.

Meromorphic solutions of

Hayman showed that for n≥9, there do not exist three nonconstant meromorphic functions f g, and h that satisfy . There are examples which show that this theorem does not hold when n = 6 and 1 ≥n≥4. In this paper we give an example which shows that this theorem does not hold when n=5.

A Generalization of the Airy Integral for f" - z n f = 0

It is well known that the Airy integral is a solution of the Airy differential equation f″ − zf = 0 and that the Airy integral is a contour integral function with special properties. We show that there exist analogous special contour integral solutions of the more general equation f″ − z n f = 0 where n is any positive integer. Related results are given

The deficiencies of meromorphic solutions of certain algebraic differential equations

We use the spread relation to prove estimates that contain the Nevanlinna deficiencies of values of meromorphic solutions of certain differential equations of the form (1.1) below. We construct examples which show that all of our estimates are sharp, and in most of these constructions we use functions which are extremal for the spread relation. Several other examples are also given to illustrate our results.

Meromorphic Solutions of a Differential Equation with Polynomial Coefficients

We give new estimates for the maximum number M of distinct meromorphic solutions and also for the maximum number L of linearly independent meromorphic solutions of the first order differential equation