Asymptotic behavior of functions with bounded boundary rotation

Abstract

Fork≧2k \geqq 2denote byVk{V_k}the class of normalized functions, analytic in the unit disc, which have boundary rotation at mostkπk\pi. Letan{a_n}be thenth Taylor coefficient off(z)∈Vkf(z) \in {V_k}. LetIλ(r,f′){I_\lambda }(r,f’)andIλ(r,f){I_\lambda }(r,f)be theλ\lambda-integral mean off′(z)f’(z)andf(z)f(z)respectively. We determine asymptotic formulas forf′(z)f’(z), and these formulas are then applied to study the behavior of|an||{a_n}|asn→∞n \to \infty, and the behavior ofIλ(r,f′){I_\lambda }(r,f’)andIλ(r,f){I_\lambda }(r,f)asr→1r \to 1.