A new property of meromorphic functions and its applications
by
Grigor Barsegian
Institute of Mathematics of Acad. of Sci. of Armenia
Coauthors: Cung Chun Yang (Hong Kong University of Science and Technology)

Let w be a meromorphic function in the complex plane. We establish a new regularity, which is the so call similarity property that qualitatively speaking shows: any finite set of polygons P_n, which are far from each other, has many w^-1-preimages z_i(P_n) whose geometric shapes are quite similar to the shapes of initial polynomials. On the other hand these similar preimages z_i(P_n) are small and close to each other. Similarity of a polygon P_n and and its curvilinear preimage z_i(P_n) is described in terms of closeness of some metric and angular proportions taken to P_n and z_i(P_n). The obtained similarity property generalizes the proximity property of a-points of meromorphic functions [1-2] which, in turn, gives an addition to the main conclusions of value distribution theory [3] : the proximity property describes locations of a-points, in addition to numbers of a-points that are studied in value distribution.

Making use this property we are able to show that there are only 15 values (with a certain geometry) such that any functions w which is meromorphic in the complex plane and has preimages of all these values lying on a finite collection of non parallel straight lines must be a rational function. Earlier similar problem conjectured by Ozawa [4] required infinite many values for deriving the same conclusion.

References

1. Barsegian G., Proximity property of a-points of meromorphic functions, Math. Sbornic, vol. 120(162), n. 1, 1983, p. 42-63.

2. Barsegian G.A. and Yang C.C., Some new generalizations in the theory of meromorphic functions and their applications. Part 1. On the magnitudes of functions on the sets of a-points of derivatives, Complex Variables Theory Appl., vol. 41, 2000, p. 293-313.

3. Nevanlinna R., Eindeutige analytische Funktionen, Springer, 1936.

4. Ozawa M., On the solution of the functional equation f(g(z))=F(z), Y, Kodai Math. Sem. Rep., v. 20, 1968, p. 305-313.

Date received: August 1, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-06.