Abstract
It is well-known that the logarithmic coefficients play an important role in the development of the theory of univalent functions. If denotes the class of functions analytic and univalent in the open unit disk , then the logarithmic coefficients of the function are defined by . In the current paper, the bounds for the logarithmic coefficients for some well-known classes like for and were estimated. Further, conjectures for the logarithmic coefficients for functions belonging to these classes are stated. For example, it is forecasted that if the function , then the logarithmic coefficients of satisfy the inequalities Equality is attained for the function , that is,
Dedicated to the memory of Professor Gabriela Kohr (1967-2020)
1. Introduction
Let denote the open unit disk in the complex plane . Let be the category of analytic functions in for which has the following representation:
Also, let be the subclass of consisting of all univalent functions in . Then, the logarithmic coefficients of the function 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 instead of . 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 . Milin [2] conjectured the inequality that implies Robertson’s conjecture [5] and hence Bieberbach’s conjecture [6], which was the well-known coefficient problem in the theory of univalent functions. De Branges [7] 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 for , it follows that
If the functions and are analytic in , the function is called to be subordinate to the function , written , if there exists a function analytic in with , , and , such that . In particular, if is univalent in , then the following equivalence relationship holds true:
Using the principle of subordination, Ma and Minda [8] introduced the classes and , where we make here the weaker assumptions that the function is analytic in the open unit disk and satisfies , such that it has a series expansion of the form
They considered the abovementioned classes as follows:
Some special subclasses of the class and play a significant role in the Geometric Function Theory because of their geometric properties.
For example, taking where , , and , we get the classes and , respectively (see also [9, 10]). The mentioned classes with the restriction reduce to the popular Janowski starlike and Janowski convex functions, respectively. By replacing and , where , we obtain the classes and of the starlike functions of order and convex functions of order, respectively. In particular, and are the class of starlike functions and of convex functions in the open unit disk , respectively. Further, by altering and , where , we get the classes and , which are the extensions of the classes and , respectively (see [11]), that is, where .
Supposing that is such that each function is of the form and is the extremal function for various problems in . Also, suppose that is such that
Then, each function is of the form and plays as extremal function for some extremal problems in the set .
Lately, Kanas et al. [12] introduced the categories and by and obtained some geometric properties in these categories. Further, the functions play as extremal functions for some issues of the families and , respectively.
Lately, several researchers have subsequently investigated same problems regarding the logarithmic coefficients and the coefficient problems [9, 13–23], to mention a few of them. For instance, the rotation of the Koebe function for each has the logarithmic coefficients , . If , then by applying the Bieberbach inequality for the first relation of (5), it follows that , 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 of satisfy the inequality and the equality is obtained for the Koebe function. For , the inequality holds but is not true for the full class , even in order of magnitude (see [24], Theorem 8.4). In 2018, some first logarithmic coefficients 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 is presumably still a concern. In [13], the authors obtained the bounds of logarithmic coefficients , , for the general class , and the bounds of the logarithmic coefficients when for the class , 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 , , of the class for and were estimated. Further, conjectures for the logarithmic coefficients for belonging to these classes are stated.
2. Main Results
First, we will obtain the bounds for of the classes and for . In this regard, the following outcomes will be employed in the key results.
Lemma 1 (see [13], Theorem 1). Let . If is convex univalent, then the logarithmic coefficients of satisfy the following inequalities: The inequalities in (18) and (19) are sharp, such that for any , there exist the function given by and the function given by , respectively, for those equalities we obtain.
Lemma 2 (see [13], Theorem 2). Let . Then, the logarithmic coefficients of satisfy the inequalities and if , and are real values, then where is given in ([26], Lemma 2) (or [9], Lemma 5), , and . The bounds (20) and (21) are sharp.
Lemma 3 (see [18], Theorem 30). If , then The first two bounds are sharp for and , respectively.
If we consider Lemma 1 with the function , then we immediately get the next result:
Theorem 4. If , then These inequalities are sharp for and , respectively.
Corollary 5. Let . Then, the logarithmic coefficients of satisfy the inequalities
Equalities in these inequalities are attained for the functions for , respectively.
Proof. For , where , in Theorem 6, we obtain the required result. Also, since it follows that these inequalities are attained for the functions for , respectively.☐
Theorem 6. Let . Then, the logarithmic coefficients of satisfy the inequalities This inequality is sharp for for the function .
Proof. If , this is equivalent to and
If we define , then , and the above subordination relation can be written as
Supposing that the function satisfies the differential equation
we will prove that is a convex univalent function in .
The function has positive real part in whenever . Therefore, using ([27], Theorem 1) for , , and , it follows that the solution of the differential equation (30) is analytic in , with for all , and
where
and all powers are considered at the principal branch, that is, .
Since is convex and is analytic with for all , using [28] (Theorem 3.2i) for , we deduce that is univalent in . Moreover, from Figure 1 made with MAPLE software, we get
and , so is a convex function. Hence, it follows that is a convex univalent function in .
Therefore, according to [28] (Theorem 3.2i), the differential subordination (29) implies
for all , and is the best dominant. Thus,
for all . Hence,
From the above relation, we get
Hence, from Lemma 1, we obtain
Therefore, for and for all , we conclude that
☐
Remark 7. If we compare the results of Corollary 5 with those of Theorem 6, then we conclude that the results of Theorem 6 are not the best possible. We conjecture that if the function , then the logarithmic coefficients of satisfy the inequalities Equality is attained for the function , that is,
Theorem 8. Let . Then, the logarithmic coefficients of satisfy the inequalities This inequality is sharp for for the function .
Proof. Letting , it follows that
Suppose that satisfies the differential equation
If we define , then the subordination (43) can be rewritten as
According to the inequality (20) of [12] (Theorem 2.3), the function is analytic with positive real part in . Therefore, using [27] (Theorem 1) for , , and , it follows that the solution of the differential equation (44) is analytic in with , , and
where
and all powers are considered at the principal branch, that is, . Moreover, from Figure 2 made with MAPLE software, we get
and . Hence, is convex in .
Since satisfies in the subordination (45), using [28] (Theorem 3.2i), we conclude that , that is,
and is the best dominant. Thus, implies , that is,
Therefore, since is convex univalent, from Lemma 1, it follows that
and we obtain the result. This completes the proof.□
Remark 9. If we compare the results of Lemma 1 with those of Theorem 8, then we conclude that the results of Theorem 8 are not the best possible. We conjecture that if the function , then the logarithmic coefficients of satisfy the inequalities Equality is attained for the function , that is,
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The fourth author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2019R1I1A3A01050861).