A class of gap series with small growth in the unit disc

L. R. Sons and Zhuan Ye

International Journal of Mathematics and Mathematical Sciences, 2002, vol. 32, 1-12

Abstract:

Let β > 0 and let α be an integer which is at least 2 . If f is an analytic function in the unit disc D which has power series representation f ( z ) = ∑ k = 0 ∞ a k z k α , lim sup k → ∞ ( log + | a k | / log k ) = α ( 1 + β ) , then the first author has proved that f is unbounded in every sector { z ∈ D : Φ − ϵ < arg z < Φ + ϵ , for ϵ > 0 } . A natural conjecture concerning these functions is that lim sup r → 1 − ( log L ( r ) / log M ( r ) ) > 0 , where L ( r ) is the minimum of | f ( z ) | on | z | = r and M ( r ) is the maximum of | f ( z ) | on | z | = r . In this paper, investigations concerning this conjecture are discussed. For example, we prove that lim sup r → 1 − ( log L ( r ) / log M ( r ) ) = 1 and lim sup r → 1 − ( L ( r ) / M ( r ) ) = 0 when a k = k α ( 1 + β ) .

Date: 2002
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:914873

DOI: 10.1155/S0161171202111136

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