Logarithmic Coefficient Bounds and Coefficient Conjectures for Classes Associated with Convex Functions

Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316-36155, Shahrood, Iran Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania Department of Applied Mathematics, College of Natural Sciences, Pukyong National University, Busan 48513, Republic of Korea


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: 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: 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 logarithmic 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 proved Bieberbach's conjecture by establishing Milin's conjecture.
Recall that we can rewrite (2) in the power series form as follows: and equating the coefficients of z n for n = 1, 2, 3, it follows that If the functions f and g are analytic in U, the function f is called to be subordinate to the function g, written f ðzÞ ≺ gðzÞ, if there exists a function w analytic in U with jwðzÞj < 1, z ∈ U, and wð0Þ = 0, such that f = g ≺ w. In particular, if g is univalent in U, then the following equivalence relationship holds true: Using the principle of subordination, Ma and Minda [8] introduced the classes S * ðφÞ and CðφÞ, where we make here the weaker assumptions that the function φ is analytic in the open unit disk U and satisfies φð0Þ = 1, such that it has a series expansion of the form They considered the abovementioned classes as follows: Some special subclasses of the class S * ðφÞ and CðφÞ play a significant role in the Geometric Function Theory because of their geometric properties.
Supposing that Ψ α,n ∈ S * ð1 + αzÞ is such that each function Ψ α,n is of the form and is the extremal function for various problems in S * ð1 + αzÞ. Also, suppose that L α,n ∈ Cð1 + αzÞ is such that Then, each function L α,n is of the form and plays as extremal function for some extremal problems in the set Cð1 + αzÞ. Lately, Kanas et al. [12] introduced the categories S T hpl ðsÞ and CV hpl ðsÞ by and obtained some geometric properties in these categories. Further, the functions 2 Journal of Function Spaces play as extremal functions for some issues of the families ST hpl ðsÞ and CV hpl ðsÞ, respectively. Lately, several researchers have subsequently investigated same problems regarding the logarithmic coefficients and the coefficient problems [9,[13][14][15][16][17][18][19][20][21][22][23], to mention a few of them. For instance, the rotation of the Koebe function kðzÞ = z ð1 − e iθ zÞ −2 for each θ ∈ ℝ has the logarithmic coefficients γ n = e iθn /n, n ≥ 1. If f ∈ S, then by applying the Bieberbach inequality for the first relation of (5), it follows that jγ 1 j ≤ 1, and using the Fekete-Szegö inequality for the second relation of (5) (see [24], Theorem 3.8) leads to It was established in ( [25], Theorem 4) that the logarithmic coefficients γ n of f ∈ S satisfy the inequality and the equality is obtained for the Koebe function. For f ∈ S * , the inequality jγ n j ≤ 1/n holds but is not true for the full class S, even in order of magnitude (see [24], Theorem 8.4).
In 2018, some first logarithmic coefficients γ n were estimated for special subclasses of close-to-convex functions in [15,20]. However, the problem of the best upper bounds for the logarithmic coefficients of univalent functions for n ≥ 3 is presumably still a concern. In [13], the authors obtained the bounds of logarithmic coefficients γ n , n ∈ ℕ, for the general class S * ðφÞ, and the bounds of the logarithmic coefficients γ n when n = 1, 2, 3 for the class KðφÞ, while the estimated bounds would generalize many of the previous outcomes.
In the present study, which is motivated essentially by the recent works [13,16], the bounds for the logarithmic coefficients γ n , n ∈ ℕ, of the class Cð1 + αzÞ for α ∈ ð0, 1 and C V hpl ð1/2Þ were estimated. Further, conjectures for the logarithmic coefficients γ n for f belonging to these classes are stated.

Main Results
First, we will obtain the bounds for γ n of the classes S * ð1 + αzÞ and Cð1 + αzÞ for α ∈ ð0, 1. In this regard, the following outcomes will be employed in the key results.
Lemma 1 (see [13], Theorem 1). Let f ∈ S * ðφÞ. If φ is convex univalent, then the logarithmic coefficients of f satisfy the following inequalities: The inequalities in (18) and (19) are sharp, such that for any n ∈ ℕ, there exist the function f n given by zf n ′ ðzÞ/f n ðzÞ = φðz n Þ and the function f given by zf ′ ðzÞ/f ðzÞ = φðzÞ, respectively, for those equalities we obtain.
If we consider Lemma 1 with the function φðzÞ ≕ 1 + αz, then we immediately get the next result: These inequalities are sharp for f = Ψ α,n and f = Ψ α,1 , respectively.

Journal of Function Spaces
Corollary 5. Let f ∈ Cð1 + αzÞ. Then, the logarithmic coefficients of f satisfy the inequalities Equalities in these inequalities are attained for the functions L α,n for n = 1, 2, 3, respectively.
Proof. If f ∈ Cð1 + αzÞ, this is equivalent to f ∈ A and If we define pðzÞ ≔ zf ′ ðzÞ/f ðzÞ, then pð0Þ = 1, and the above subordination relation can be written as Supposing that the function ψ α satisfies the differential equation we will prove that ψ α is a convex univalent function in U.
The function φ α has positive real part in U whenever α ∈ ð0, 1. Therefore, using ( [27], Theorem 1) for β = 1, γ = 0, and c = 1, it follows that the solution ψ α of the differential equation (30) is analytic in U, with Re ψ α ðzÞ > 0 for all z ∈ U, and where and all powers are considered at the principal branch, that is, log 1 = 0.