On the zero sets of certain entire functions

Eremenko, Alexandre; Rubel, L.

doi:10.1090/S0002-9939-96-03294-7

On the zero sets of certain entire functions
HTML articles powered by AMS MathViewer

by Alexandre Eremenko and L. A. Rubel PDF

Proc. Amer. Math. Soc. 124 (1996), 2401-2404 Request permission

Abstract:

We consider the class $\mathbf B$ of entire functions of the form \[ f=\sum p_j\exp g_j,\] where $p_j$ are polynomials and $g_j$ are entire functions. We prove that the zero-set of such an $f$, if infinite, cannot be contained in a ray. But for every region containing the positive ray there is an example of $f\in \mathbf B$ with infinite zero-set which is contained in this region.

References

H. Cartan, Sur les zéros des combinations linéaires de p fonctions holomorphes données, Mathematica (Cluj), 7 (1933), 5-31.
Albert Edrei and Wolfgang H. J. Fuchs, On the growth of meromorphic functions with several deficient values, Trans. Amer. Math. Soc. 93 (1959), 292–328. MR 109887, DOI 10.1090/S0002-9947-1959-0109887-4
D. Gaĭer, Lektsii po teorii approksimatsii v kompleksnoĭ oblasti, “Mir”, Moscow, 1986 (Russian). Translated from the German by L. M. Kartashov; Translation edited and with a preface by V. I. Belyĭ and P. M. Tamrazov. MR 894919
C. Ward Henson, Lee A. Rubel, and Michael F. Singer, Algebraic properties of the ring of general exponential polynomials, Complex Variables Theory Appl. 13 (1989), no. 1-2, 1–20. MR 1029352, DOI 10.1080/17476938908814374
Serge Lang, Introduction to complex hyperbolic spaces, Springer-Verlag, New York, 1987. MR 886677, DOI 10.1007/978-1-4757-1945-1
Joseph Miles, On entire functions of infinite order with radially distributed zeros, Pacific J. Math. 81 (1979), no. 1, 131–157. MR 543739, DOI 10.2140/pjm.1979.81.131