Abstract
The present study’s intention is to produce exact estimations of some problems involving logarithmic coefficients for functions belonging to the considered subcollection of the bounded turning class. Furthermore, for the class , we look into the accurate bounds of the Zalcman inequality, Fekete-Szegö inequality along with and . Importantly, all of these bounds are shown to be sharp.
1. Introduction and Definitions
To properly understand the findings provided in the article, certain important literature on Geometric Function Theory must first be discussed. In this regard, the letters and stand for the normalized univalent functions class and the normalized holomorphic (or analytic) functions class, respectively. These primary notions are defined in the region by where symbolizes the holomorphic functions class, and
The following formula defines the logarithmic coefficients of that belong to
In many estimations, these coefficients provide a significant contribution to the concept of univalent functions. In 1985, De Branges [1] proved that and equality will be achieved if has the form for some In its most comprehensive version, this inequality offers the famous Bieberbach-Robertson-Milin conjectures regarding Taylor coefficients of . We refer to [2–4] for further details on the proof of De Branges’ finding. By considering the logarithmic coefficients, Kayumov [5] was able to prove Brennan’s conjecture for conformal mappings in 2005. For your reference, we mention a few works that have made major contributions to the research of the logarithmic coefficients. Andreev and Duren [6], Alimohammadi et al. [7], Deng [8], Roth [9], Ye [10], Obradović et al. [11], and finally the work of Girela [12] are the major contributions to the study of logarithmic coefficients for different subclasses of holomorphic univalent functions.
As stated in the definition, it is simple to determine that for , the logarithmic coefficients are computed by
For given , , and with the series expansion (1), the Hankel determinant is represented by
It was defined by Pommerenke [13, 14]. This determinant has indeed been investigated for a number of univalent function subclasses. In specific, the sharp estimate of the functional for the sets (convex functions), (starlike functions), and (bounded turning functions) has been effectively established in [15, 16]. Later, numerous scholars published their findings on the upper bounds of for various subcollections of holomorphic functions; see [17–23]. However, for the class of close-to-convex functions, the exact estimation of this determinant is yet unknown [24].
Analogous to the determinant mentioned above, Kowalczyk and Lecko [25, 26] considered to examine the following determinant with entries from logarithmic coefficients of
It is observed that
For the given functions the subordination between and (mathematically written as ), if we get a Schwarz function with and for in a way such that hold true. Additionally, the following relation applies if in is univalent: if and only if
In 1992, Ma and Minda [27] developed a consolidated version of the collection by using the principle of subordination, and the following is a description of it: where the univalent function satisfies
The area is also symmetric about -axis and has a star-shaped form around the point . In recent years, a wide variety of the collection ’s subfamilies have been looked into as particular alternatives for the class . As an illustration: (i) with and (see [28])(ii) (see [29]), and (see [30, 31])(iii) (see [32]), and (see [33, 34])(iv) (see [35]), and (see [36])(v) (see [37, 38])
In [39], Cho et al. developed a novel subfamily of starlike function described by
From the definition of the family the authors [39] deduced that for some By substituting in (17), we acquire the function which acts as the extremal function in a variety of -family problems. In [40], the authors defined the following subfamily of holomorphic functions by using (18):
Our primary objective in the current paper is to compute the problems involving the sharp logarithmic coefficients for the class of bounded turning functions connected to an eight-shaped domain. The sharp bounds of the Zalcman inequality, the Fekete-Szegö type inequality, along with the determinants and for the family are found using logarithmic coefficient entries.
2. Preliminary Lemmas
We must first create the class in the below set-builder form in order to declare the Lemmas that are employed in our primary findings:
That is, if , then it has the series representation
Lemma 1 (see [41]). Let and has the series form (22). Then for
Lemma 2. If and has the expansion (22), then and if and then Also, The inequalities (27), (28) and (29) are taken from [42, 43], and [44], respectively.
Lemma 3 (see [45]). Let , and satify the inequalities and If has the form (22), then
3. Coefficient Inequalities for the Class
Theorem 4. If and has the series representation (1), then These bounds are sharp and can be obtained from the following extremal functions
Proof. Let Consequently, (20) may be expressed using the Schwarz function as The Schwarz function may be used to express it if as follows equivalently, From (1), we obtain By simplification and using the series expansion of (36), we get Comparing (37) and (38), we obtain Putting (42) in (5), (6), (7), and (8), we obtain For using (27), in (43), we obtain For putting (29) in (44), we obtain For we can rewrite (45) as Using (28) we get For we can rewrite (46) as Comparing the right side of (51) with where It follows that Using (30) we deduce that
Theorem 5. If has the series form (1) and belongs to then Equality will be attained by using (5), (6), and
Proof. From (43) to (44), we get Using (29), we have After the simplification, we get
Theorem 6. If has the series expansion (1) and belongs to then Equality can be attained by applying (5), (6), (7), and
Proof. From (43), (44), and (45), we obtain After the simplification, we obtain Using (28), we have
Theorem 7. If has given by (1), then This result is sharp and equality can be achieved by applying (6), (8), and
Proof. From (44) to (46), we obtain After the simplification, we obtain Comparing the right side of (69)with where It follows that Using (30) we deduce that
4. Hankel Determinant with Logarithmic Coefficients
Theorem 8. Let and be of the form (1). Then The above stated result is sharp. Equality can be attained with the use of (5), (6), (7), and
Proof. Employing (43), (44), and (45), we obtain Using (23) and (24) along with the assumption that we get Applying triangle inequality and assuming and also setting , we have A little exercise can verify that in , and this implies Thus, by choosing we achieve Now, since we see that is a decreasing function, and so its maximum value appears at the lowest point , which is
Theorem 9. If and has the form (1), then The inequality is sharp and can be obtained by using (6), (7), (8), and
Proof. The determinant can be written as
Putting (44), (45), and (46), with , we obtain
Let in (23), (24), and (25). Now, applying the simplest version of the given lemma, we get
Putting the above expressions in (84), we get,
Since where and
Now, by the virtue of , and we get
where
with
To illustrate the sharp bounds of the given problem, we must maximize in the closed cuboid .
(1)Interior points of cuboid Let us choose Then simple calculation yields
Putting we obtain
If is a critical point, then , and it is applicable only if
To check critical points existence, we must find solutions that fulfill both constraints (94) and (95).
Let As for all it is evident that is decreasing in Hence It is easy to showcase that the inequality (94) does not hold in this scenario for all . As a result, does not have a critical point in . Assume a critical point of exists inside the interior of the cuboid , it must unquestionably fulfil that
From the arguments above, it is undeniable that and Now let us establish that For by invoking and , it is not hard to observe that
Therefore, we have
Obviously, it can be seen that
Since for we obtain that for , and thus, it follows that
Therefore, we have
It is easy to be calculated that attains its maximum value at Thus, we have
Hence, This implies that is less than at all the critical points in the interior of Therefore, has no optimal solution in the interior of (2)Interior of all the six faces of cuboid (i)On the face yieldsDifferentiating with respect to we have
Thus, has no critical point in the interval (ii)On the face becomes(iii)On the face reduces toDifferentiating partially with respect to , we have
Solving we obtain
For the given range of should belong to which is possible only if Also derivative of partially with respect to is
Putting the value of in (108), with and simplifying, we obtain
A calculation gives the solution of (109) in the interval , that is, . Thus, has no optimal point in the interval (iv)On the face becomesThen
By setting , we get the critical point at which attains its maximum value, which is given below
(v)On the face yieldsA numerical computation shows that the solution for the system of equations
does not exists in the interval Hence, has no optimal solution in the interval (vi)On the face reduces toAs in the above case, we conclude the same result for the face that is, system of equations
has no solution in the interval (3)On the edges of cuboid (i)On the edge and reduces toIt follows that
We see that for the critical point at which obtain its maximum value, which is given by
(ii)On the edge and becomesDifferentiating with respect to we have
We know that in follows that is decreasing over Therefore, gets its maxima at Hence
(iii)On the edge and reduces toNoting that in shows that is increasing over Thus, gets its maxima at Thus, we have
(iv)On the edges and Since is free of therefore
Then
By putting we obtain the critical point at which attains its maximum value, which is given by
(v)On the edge and becomes(vi)On the edge reduces to is independent of and ; therefore
(vii)On the edge and takes the formIt is clear that
We see that in shows that is decreasing over Thus, gets its maxima at Hence, we have
(viii)On the edge and yieldsIt follows that
By taking , we obtain the critical point at which attains its maximum value, which is given by
Hence, from the above cases we deduce that
From (89)we have
If then sharp bound for this Hankel determinant is determined by
Thus, we have completed the proof.
5. Conclusion
In our current investigation, we have considered a class of bounded turning functions associated with an eight-shaped domain. For such a class, we studied some interesting problems involving logarithmic coefficients. The Zalcman inequality, the Fekete-Szegö inequality, and the determinants and for the family have been studied here in this article. All the obtained results are proven to be the best possible.
Data Availability
The numerical data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work under Grant Code number: 22UQU4310396DSR23.