Sharp estimate of the fifth coefficients for the class U(λ)

Let f be function that is analytic in the unit disk D = {z : z| 1}, normalized such that f(0) = f′(0) − 1 = 0, i.e., of type f(z) = z + ∑∞n=2 an zn. If additionally, |(z/f(z))² f′(z) − 1| λ (z ∈ D), then f belongs to the class U(λ), 0 λ ≤ 1. In this paper we prove sharp upper bound of the modulus of the fifth coefficient of f from U(λ) satisfying f(z)/z ≺ 1/(1 + z)(1 + λz) , (”≺” is the usual subordination) in the case when 0.400436 . . . ≤ λ ≤ 1.