Logarithmic Coefficient Bounds and Coefficient Conjectures for Classes Associated with Convex Functions. (14th June 2021) - Alimohammadi, Davood; Analouei Adegani, Ebrahim; Bulboacă, Teodor; Cho, Nak Eun
Abstract : 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 z = z + ∑ n = 2 ∞ a n z n analytic and univalent in the open unit disk U, then the logarithmic coefficients γ n f of the function f ∈ S are defined by log f z / z = 2 ∑ n = 1 ∞ γ n f z n . In the current paper, the bounds for the logarithmic coefficients γ n for some well-known classes like C 1 + α z for α ∈ 0, 1 and C V hpl 1 / 2 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 ∈ C 1 + α z, then the logarithmic coefficients of f satisfy the inequalities γ n ≤ α / 2 n n + 1, n ∈ ℕ . Equality is attained for the function L α, n, that is, log L α, n z / z = 2 ∑ n = 1 ∞ γ n L α, n z n = α / n n + 1 z n + ⋯, z ∈ U .