COEFFICIENT PROBLEMS ON THE CLASS U(λ) - Ponnusamy Saminathan; Wirths Karl-Joachim (Проблемы анализа, 2018)
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.