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Growth along a ray of a subharmonic function, having a mass distributed on the negative axis

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Abstract

Let U be a subharmonic function in C with a Riesz mass Μ, distributed on the negative semiaxis without some neighborhood of zero, let ρ and λ be its order and lower order, and let B(r, U) be the maximum of U(z) for ¦z¦=r. Estimates are obtained for the measure of sets of those values of r ⩾ 0 for which certain inequalities hold. The following result is typical. LetE = {r:u(re lθ)−cosθσB<(r,U) > 0}. If ρ < σ < 1, ¦θ¦=π., then the lower logarithmic density of the set E is at least 1 − ρ/σ. If λ < σ> 1,¦θ¦ ⩽ π., then the upper logarithmic density of the set E is at least 1 − λ/σ.

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Authors
  1. A. A. Gol'dberg
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  2. O. P. Sokolovskaya
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Additional information

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 31–38, 1988.

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Gol'dberg, A.A., Sokolovskaya, O.P. Growth along a ray of a subharmonic function, having a mass distributed on the negative axis. J Math Sci 49, 1258–1262 (1990). https://doi.org/10.1007/BF02209169

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  • DOI: https://doi.org/10.1007/BF02209169

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