A class of gap series with small growth in the unit disc. (29th September 2002)

• Record Type: Journal Article
• Authors: Sons, L. R.; Ye, Zhuan
• Is Part Of: International journal of mathematics and mathematical sciences. Volume 32:Number 1(2002)
• Page Start: 29
• Page End: 40
• Publication Date: 2002-09-29
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1155/S0161171202111136
• Languages: English
• ISSNs: 0161-1712

Abstract

Abstract : Letβ > 0 and letα be an integer which is at least2 . Iff is an analytic function in the unit discD which has power series representationf ( 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 thatf is unbounded in every sector{ z ∈ D : Φ − ϵ < arg z < Φ + ϵ, forϵ > 0 } . A natural conjecture concerning these functions is thatlim sup r → 1 − ( log L ( r ) / log M ( r ) ) > 0, whereL ( r ) is the minimum of| f ( z ) | on| z | = r andM ( r ) is the maximum of| f ( z ) | on| z | = r . In this paper, investigations concerning this conjecture are discussed. For example, we prove thatlim sup r → 1 − ( log L ( r ) / log M ( r ) ) = 1 andlim sup r → 1 − ( L ( r ) / M ( r ) ) = 0 whena k = k α ( 1 + β ) .