Abstract
The logarithmic coefficients of a normalized analytic functions are defined by . For certain close-to-convex functions , Cho et al. (on the third logarithmic coefficient in some subclasses of close-to-convex functions) has obtained the upper bound of the third logarithmic coefficient when the second coefficient is real. In the present paper, the upper bound of the third logarithmic coefficient is computed with no restriction on the second coefficient .
1. Introduction and Preliminaries
Let be the open unit disk in the complex plane and let be the set of all analytic normalized functions of the form
Let be its subclass consisting of functions that are univalent in . Given a function , the coefficients are defined by
For example (see Figure 1), for the Koebe function given by , the logarithmic coefficients are as follows
The Milin conjecture ([1] and ([2] p. 155)) gives an inequality satisfied by the logarithmic coefficients. For , the logarithmic coefficients satisfy
The Milin conjecture was confirmed (e.g., ([2] p. 37), by Branges [3] and implies the famous Bieberbach conjecture that for . Sharp estimates for the class are known only for the first two coefficients:
Note that Obradović and Tuneski [4] obtained an upper bound of for the class . The problem of estimating the modulus of the first three logarithmic coefficients is significantly studied for the subclasses of , and in some cases, sharp bounds are obtained. For instance, sharp estimates for the class of starlike functions are given by the inequality holds for ([5], p. 42).
Furthermore, for the class of strongly starlike function of the order , , it holds that [6]. The bounds of for functions in subclasses of have been widely studied in recent years. Sharp estimates for different subclasses are given in [6, 7] and ([5], p. 116) and [8], respectively, while nonsharp estimates for the class of Bazilevic and close-to-convex are given in [9–11], respectively.
Let be the subclasses of satisfying, respectively, the next conditions:
Note that each class defined above is the subclass of the well-known class of close-to-convex functions; consequently, families , , contain only univalent functions ([2], Vol. II, p. 2). The sharp bounds of and partial results for of the subclasses of were determined by Pranav Kumar and Vasudevarao [12].
Moreover, Cho et al. [13] computed the sharp upper bounds for the third logarithmic coefficient of when is a real number. Differentiating (1) and comparing the coefficients with (2), we get , , and
The main aim of this paper is to determine the upper bound of the third logarithmic coefficient in the general case of . The following lemma is needed to prove our main results.
Lemma 1 (see [14]). Let be a Schwarz function. Then
2. Main Results
Our main result is as follows:
Theorem 1. Let . Then
Proof. Since , and for analytic function in with satisfying the formulaWe obtainThen, by using (10) along with (11) leads toFrom (7) and (12), we obtainIn view of Lemma 1, we attainwhereThe systemhas a unique solution withThe maximum value of is obtained when is a point on the boundary of . In view of this, we haveandUsing (14) and (17)–(19), we conclude the following outcome:This completes the proof.
Remark 1. If , where is a real number, then we get the result in [13]
Theorem 2. Let . Then
Proof. Since , then there exists an analytic function in with andThe coefficients can be determined by comparing the information in (11) and (23)From (7) and (24), we have the following conclusion:Moreover, according to Lemma 1, we get the following inequality:whereFrom the system,only one solution lies in the interior of , whereandOn the boundary of , we have the next propertyConsequently, (26), (30), and (31) yield
Remark 2. If , where is a real number, then [13]
Theorem 3. Let . Then
Proof. Let and an analytic function in with such thatSubstituting (11) into (35), we haveBy using (7) and (36), we obtainAccording to Lemma 1, we conclude thatwhereThe systemadmits a unique solution in the interior of such thatOn the boundary of , the following cases are observed:andEquations (38), (41)–(43) show that
Remark 3. Let , where is a real number. Then [13]
Data Availability
No data were used in this study.
Disclosure
The author would like to declare that a preprint of this article has previously been published in [15].
Conflicts of Interest
The author declares that there are no conflicts of interest.