Uniformly convex functions II (original) (raw)
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It is well-known that the logarithmic coefficients play an important role in the development of the theory of univalent functions. If S denotes the class of functions f ðzÞ = z + ∑ ∞ n=2 a n z n analytic and univalent in the open unit disk U, then the logarithmic coefficients γ n ðf Þ of the function f ∈ S are defined by log ð f ðzÞ/zÞ = 2∑ ∞ n=1 γ n ð f Þz n. In the current paper, the bounds for the logarithmic coefficients γ n for some well-known classes like Cð1 + αzÞ for α ∈ ð0, 1 and CV hpl ð1/2Þ were estimated. Further, conjectures for the logarithmic coefficients γ n for functions f belonging to these classes are stated. For example, it is forecasted that if the function f ∈ Cð1 + αzÞ, then the logarithmic coefficients of f satisfy the inequalities jγ n j ≤ α/ð2nðn + 1ÞÞ, n ∈ ℕ: Equality is attained for the function L α,n , that is, log ðL α,n ðzÞ/zÞ = 2∑ ∞ n=1 γ n ðL α,n Þz n = ðα/nðn + 1ÞÞz n + ⋯,z ∈ U: