Annales Academić Scientiarum Fennicć
Mathematica
Volumen 34, 2009, 379-386

A TECHNIQUE OF BEURLING FOR LOWER ORDER

P.C. Fenton and John Rossi

University of Otago, Department of Mathematics and Statistics
P.O. Box 56, Dunedin, New Zealand; pfenton 'at' maths.otago.ac.nz

Virginia Tech, Department of Mathematics
Blacksburg, VA 24061-0123, U.S.A.; rossij 'at' vt.edu

Abstract. Suppose that \phi = u - v, where u and v are subharmonic in the plane, with u nonconstant. Suppose also that liminf_r\to\infty(log(B(r,u) + B(r,v)) / log r \le \lambda, for some \lambda satisfying 0 < \lambda < 1/2, and that the deficiency \delta of \phi satisfies 0 \le 1 - \delta < cos \pi\lambda. Given \sigma satisfying \lambda < \sigma < 1/2 and 0 \le 1 - \delta < cos \pi\sigma, we have

A(r,\phi) / B(r,\phi) > \kappa = \kappa(\sigma,\delta) := (cos\ pi\sigma - (1 - \delta)) / (1 - (1 - \delta) cos \pi\sigma)

for all r in a set of upper logarithmic density at least 1 - \lambda / \sigma. Here A(r,\phi) = inf_|z|=r \phi(z) and B(r,\phi) = sup_|z|=r \phi(z).

2000 Mathematics Subject Classification: Primary 30D15.

Key words: \delta-subharmonic, lower order, deficiency.

Reference to this article: P.C. Fenton and J. Rossi: A technique of Beurling for lower order. Ann. Acad. Sci. Fenn. Math. 34 (2009), 379-386.

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