Annales Academić Scientiarum Fennicć
Mathematica
Volumen 34, 2009, 69-108

ON A PROBLEM OF NEVANLINNA

Joseph Miles

University of Illinois, Department of Mathematics
1409 West Green Street, Urbana, IL 61801, U.S.A.; joe 'at' math.uiuc.edu

Abstract. If f is a meromorphic function on the plane, let

K(f) = lim sup_r\to\infty (N(r,0,f) + N(r,\infty,f)) / T(r,f),

where we use standard functionals from Nevanlinna theory. It has long been conjectured for all meromorphic functions of finite nonintegral order \rho that K(f)\ge K(L_\rho), where the entire function L_\rho is the canonical product with positive zeros satisfying n(r,0,L_\rho) = [r^\rho]. This conjecture has been established only for \rho < 1. We show the existence of \rho₀ > 1 such that if 1 < \rho < \rho₀ then K(f)\ge K(L_\rho) for all meromorphic f of order \rho satisfying N(r,0,f) + N(r,\infty,f) ~ cr^\rho for some c > 0.

2000 Mathematics Subject Classification: Primary 30D35.

Key words: Meromorphic function, Nevanlinna theory, Fourier series.

Reference to this article: J. Miles: On a problem of Nevanlinna. Ann. Acad. Sci. Fenn. Math. 34 (2009), 69-108.

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