A generalization of trigonometric convexity and its relation to positive harmonic functions in homogeneous domains

Details A generalization of trigonometric convexity and its relation to positive harmonic functions in homogeneous domains A generalization of trigonometric convexity and its relation to positive harmonic functions in homogeneous domains

2004-07-18

arxiv.org/abs/math/0407315v1

Mathematics

Complex Variables

41 pages; to appear in Journal d'Analyse Mathematique

Scientific paper

We consider functions which are subfunctions with respect to the differential operator $$L_\rho = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + 2\rho \frac{\partial}{\partial x} + \rho^2 $$ and are doubly periodic in the plane. These functions play an important role in describing the asymptotic behavior of entire and subharmonic functions of finite order. In studying their properties we are led to problems concerning the uniqueness of Martin functions and the critical value for the parameter $\rho$ in the homogeneous boundary problem for the operator $L_\rho$ in a domain on the torus.

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