COEFFICIENT PROBLEMS ON THE CLASS U(λ)

• Main Authors: Ponnusamy Saminathan, Wirths Karl-Joachim
• Format: Article
• Language: English
• Published: Petrozavodsk State University 2018-06-01
• Series: Проблемы анализа
• Subjects: Univalent function ◆ subordination ◆ Julia’s lemma ◆ Schwarz’ lemma
• Online Access: http://issuesofanalysis.petrsu.ru/article/genpdf.php?id=4730&lang=ru

For 0 < λ ≤ 1, let U(λ) denote the family of functions f(z)=... analytic in the unit disk D satisfying the condition |...| < λ in D. Although functions in this family are known to be univalent in D, the coefficient conjecture about an for n ≥ 5 remains an open problem. In this article, we shall first present a non-sharp bound for |an|. Some members of the family U(λ) are given by z/f(z) = 1 - (1 + λ)φ(z) + λ(φ(z))^2 with φ(z) = e^iθ*z, that solve many extremal problems in U(λ). Secondly, we shall consider the following question: Do there exist functions φ analytic in D with |φ(z)| < 1 that are not of the form φ(z) = e^iθ*z for which the corresponding functions f of the above form are members of the family U(λ)? Finally, we shall solve the second coefficient (a_2) problem in an explicit form for f ∈ U(λ) of the form f(z) = z/... , where ω is analytic in D such that |ω(z)| ≤ 1 and ω(0) = a, where a ∈ D.