Research Article Logarithmic Coefficient Bounds and Coefficient Conjectures for Classes Associated with Convex Functions Davood Alimohammadi , 1 Ebrahim Analouei Adegani , 2 Teodor Bulboacă , 3 and Nak Eun Cho 4 1 Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran 2 Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316-36155, Shahrood, Iran 3 Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania 4 Department of Applied Mathematics, College of Natural Sciences, Pukyong National University, Busan 48513, Republic of Korea Correspondence should be addressed to Davood Alimohammadi; d-alimohammadi@araku.ac.ir Received 16 December 2020; Accepted 29 May 2021; Published 14 June 2021 Academic Editor: John R. Akeroyd Copyright © 2021 Davood Alimohammadi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 ð f 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 C ð1+ 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 ∈ C ð1+ 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: Dedicated to the memory of Professor Gabriela Kohr (1967-2020) 1. Introduction Let U ≔ fz ∈ ℂ : jzj <1g denote the open unit disk in the complex plane ℂ. Let A be the category of analytic functions f in U for which f has the following representation: fz ðÞ = z + 〠 ∞ n=2 a n z n , z ∈ U: ð1Þ Also, let S be the subclass of A consisting of all univalent functions in U. Then, the logarithmic coefficients γ n of the function f ∈ S are defined with the aid of the following series expansion: log fz ðÞ z =2 〠 ∞ n=1 γ n f ðÞz n , z ∈ U: ð2Þ These coefficients play an important role for different estimates in the theory of univalent functions, and note that we use γ n instead of γ n ð f Þ. Kayumov [1] solved Brennan’s conjecture for conformal mappings with the help of studying the logarithmic coefficients. The significance of the logarith- mic coefficients follows from Lebedev-Milin inequalities ([2], chapter 2; see also [3, 4]), where estimates of the logarithmic coefficients were applied to obtain bounds on the coefficients of f . Milin [2] conjectured the inequality 〠 n m=1 〠 m k=1 k γ k j j 2 − 1 k   ≤ 0, n = 1, 2, 3, ⋯, ð3Þ that implies Robertson’s conjecture [5] and hence Bieber- bach’s conjecture [6], which was the well-known coefficient problem in the theory of univalent functions. De Branges Hindawi Journal of Function Spaces Volume 2021, Article ID 6690027, 7 pages https://doi.org/10.1155/2021/6690027