1. Introduction, Definitions and Preliminaries
For a better understanding of the work studied in this article, we have to provide certain elementary geometric function theory literature. In this regard, we first express the classes of normalized analytic and univalent functions by the letters and , respectively. These classes are defined in the following set-builder form by
(1)
and where stands for the set of analytic functions in the region The set was developed by Köebe [1] in 1907, and it has become a key component of advanced study in this subject. Later in 1916, Bieberbach [2] conjectured the coefficient estimate for the class and proved it for the second coefficient. The proof of this conjecture attracted researchers, whose work developed this field immensely. In 1985, de-Branges [3] proved this famous conjecture. From 1916 to 1985, many of the world’s most distinguished scholars sought to prove or disprove this claim. As a result, they investigated a number of subfamilies of the class of univalent functions that are associated with various image domains. The most fundamental and significant subclasses of the set are the families of starlike and convex functions, represented by and , respectively.It is worth noting that Aleman and Constantin [4] recently gave a beautiful interaction between univalent function theory and fluid dynamics. In fact, they demonstrated a simple method for how to use a univalent harmonic map to obtain explicit solutions of incompressible two-dimensional Euler equations.
For the given functions we say , if an analytic function v exists in with the restrictions and such that . If in is univalent, then we have the following relationship given by
In 1992, Ma and Minda [5] presented a unified version of the class using subordination terminology. They introduce the defined by
(2)
where is a univalent function with and In addition, the region is star-shaped about the point and is symmetric along the real line axis. They focused on distortion, growth, and covering theorems, among other interesting properties of functions in this class. Later in 2007, Rosihan et al. [6] determined the sharp bounds of problems involving coefficients for a generalized class of Ma-Minda type starlike functions. The class unifies various sub-families of starlike functions, which are attained by an appropriate choice of . For instance:(i).. By choosing the function
we achieve the class which was studied in [7]. The above described class, with the limitation , represents the class of Janowski starlike functions investigated in [8]. The special case by taking and with leads to the class of starlike function of order(ii).. The below listed class
for was introduced as the collection of strongly starlike functions of order investigated in [9].(iii).. In [10], Sharma et al. discussed the class defined by
Geometrically, it is a subclass of functions with
contained in the cardioid domain given by(iv).. By setting we attain the class of starlike functions connected with the eight-shaped domain which was introduced by Cho et al. [11]. Moreover, the below mentioned classes
were analyzed, respectively, by Raza and Bano [12] and Alotaibi et al. [13].(v).. By picking we obtain the class
which was established by Ullah et al. [14]. Moreover, they examined the radii results for the class Further, in [15] the authors computed third-order Hankel determinant sharp bounds for this class.
Finding bounds for the function coefficients in a given collection has been one of the most fundamental problems in geometric function theory since it impacts geometric features. The constraint on the second coefficient, for example, provides the growth and distortion features. The general form of the Hankel determinant for the function was explored by Pommerenke [16,17] in the form of
In fact, the determinants listed below are referred to as first-, second-, and third-order Hankel determinants, respectively.
(3)
(4)
(5)
Only a few articles on the Hankel determinant for the class can be found in the literature. The earliest recorded sharp inequality for is stated by
This outcome is due to Hayman [18]. Likewise, for the same set , the following bounds were derived in [19]:
andThe problem of determining the sharp inequalities of Hankel determinants for a certain set of functions attracted the minds of many experts. Janteng et al. [20,21] computed the sharp bound of for the sub-families and where and are the sets of convex and bounded turning functions. These results are based on estimations provided by
For the following two families and the experts [22,23] obtained that is bounded by and , respectively. This problem was also investigated for different families of bi-univalent functions in [24,25,26,27,28,29].
The formulae in (3)–(5) make it quite evident that calculating the bound for is significantly more difficult than calculating the bound for Babalola [30] was the first mathematician who studied third-order Hankel determinant for the and families. Though after the Babalola’s article, several papers appeared on obtaining the bounds of the determinant for various subclasses of analytic functions. However, Zaprawa’s article [31] gained the attention of the readers in which he enhanced Babalola’s conclusions by employing a new approach to demonstrate that
In addition, he points out that such bounds are not sharp. Later, in 2018, Kwon et al. [32] enhanced the Zaprawa inequality for by achieving , and Zaprawa et al. [33] polished this bound even further in 2021 by proving that for Moreover, for q-starlike type functions classes, such problems were determined in [34]. Furthermore, the non-sharp bounds of this determinant for the sets and were also computed in the articles [35,36], respectively. They achieved
The sharp bounds of the determinant have been sought by many experts, but none have succeeded. Eventually, in 2018, Kowalczyk et al. [37] and Lecko et al. [38] achieved the following sharp bounds of for the sets and :
Barukab et al. [39], in the year 2021, computed the sharp bounds of for functions of bounded turning set related with the petal-shaped domain. Later at the end of 2021, Ullah et al. [15] and Wang et al. [40] contributed the following sharp bounds of this determinant:
where the family is given byThe interested readers can look at the work of Srivastava et al. [41] for further contributions in this area. They successfully obtained the bounds of the fourth-order Hankel determinant for various analytic and univalent functions.
In [42], Gandhi introduced a subclass of starlike functions defined by
For functions belonging to this class, it means that lie in a three-leaf-shaped region in the right-half plane. From the definition of the family the authors [42] deduced that:
(6)
for some By substituting in (6), we acquire the function(7)
Similar to the definition of , we now define a new subfamily of bounded turning functions by the following set builder notation:
(8)
For , it can be noted that is a subclass of functions satisfying the condition
In [43], Shi et al. gave some coefficient estimates on functions belonging to the class . However, the bound that they obtained for the third Hankel determinant is not sharp. In the current paper, we aim to prove some sharp bounds on the coefficient problems associated with and using a new method. The main results are organized as follows. The first part ais coefficient problems connected with the newly defined subclass of bounded turning functions. In the second part, we give some sharp bounds of third Hankel determinant for the functions in the class which improve the known results.
2. A Set of Lemmas
Before stating the results that are applied in the main contributions, we define the class in terms of a set-builder notation:
where the function q has the below series form:(9)
The subsequent Lemma is essential for the proofs of our main findings. It includes the well-known formula [44], the formula credited to Libera and Złotkiewicz [45], and the formula proven in [46].
Let be in the form of (9). Then, for we have
(10)
(11)
(12)
If has the form of (9), then
(13)
and(14)
Moreover, if with we have
(15)
The inequalities (13)–(15) in the last Lemma are taken from [44,47] and [35,36], respectively.
([48]). Let and a satisfy the inequalities and
(16)
If is of the form (9), then
3. Coefficient Inequalities and Second Hankel Determinant for the Function Class
If has the series expansion (1), then
(17)
(18)
(19)
(20)
These bounds are sharp with the extremal functions given by
Let Then from the definition, there exists a Schwarz function such that
Suppose that be described in terms of the Schwarz function as
(21)
Equivalently, we have
(22)
From (1), we obtain
(23)
Using the series expansion of (22), we obtain
(24)
Comparing (23) and (24), we find that
(25)
(26)
(27)
(28)
The inequalities on , and follow directly by using Lemma 2. For we can rewrite (28) as
(29)
whereIt can be easily verified that and
Therefore, from Lemma 3 we have
The proof of Theorem 1 is thus completed.
□
Let and be the form of (1). Then the sharp bound of the Fekete–Szegö inequality is
By employing (25) and (26), we have
An application of Lemma 2 leads to the desired result. The inequality is sharp with the extremal function given by
□
If has the form (1), then
This result is sharp.
Using (25)–(27), we have
Let and . It can seen that , and
Applying Lemma 2, we obtain the inequality in Theorem 3. This result is sharp with the extremal function given by
□
The second-order Hankel determinant for will be estimated next.
If then
The result is sharp.
By the virtue of (25)–(27), we have
Using (10) and (11) to express and in terms of and in , we obtain
Applying the triangle inequality and using , we have
It is an easy task to illustrate that for . This means that Thus
By observing that for , we see that . Thus, we have
Equality is attained by the function given by
□
4. Results on the Third Hankel Determinant of Functions
Now we study the determinant for .
If has the series expansion (1), then
The bound is sharp.
Let and put the values of ’s from (25)–(28) into (5), we obtain that
(30)
For some , by substituting in (10)–(12), we have
Putting the above expressions in (30) yields to
By virtue of , we see that
whereBy setting and utilizing the fact we obtain
(31)
where withNow, we have to maximize in the closed cuboid For this, we have to discuss the maximum values of in the interior of , in the interior of its 6 faces and on its 12 edges.
1. Interior points of cuboid:
Let and differentiating partially with respect to y, we have
Plugging we obtain
If is a critical point within then which is only achievable if
(32)
and(33)
Now, we must find solutions that meet both inequality (32) and (33) for the existence of critical points.
Let As in , it is noted that is decreasing over Hence and an easy calculation indicates that (32) is impossible for all ,Ṫhus there are no critical points of G in .
Suppose that there is a critical point of G existing in the interior of cuboid . Clearly it must satisfy that . From the above discussion, it can also be known that and . In the following, we will prove that in this situation. For , by invoking and , it is not hard to observe that
Therefore, we have
Obviously, it can be seen that
andSince for , we obtain that for , and thus it follows that
This implies that
It is easy to calculate that attains its maximum value 25,311.25 at . Thus, we have
Hence 43,200. This implies that G is less than 43,200 at all the critical points in the interior of . Therefore, G has no optimal solution in the interior of .
2. Interior of all the six faces of cuboid:
(i) On the face , becomes to
Differentiating partially with respect to we obtain
It is easy to see that has no critical point in the interval
(ii) On the face , yields
(iii) On the face , becomes
Differentiating partially with respect to y, we know that
Also derivative of partially with respect to c is
By using Newton’s methods for the system of nonlinear equations in Maple, we have found no solution to the system of equations in the interval That is, has no optimal solution in
(iv) On the face , takes the form
Then
Putting and solving we obtain Thus we have
(v) On the face , yields
Now differentiating partially with respect to then with respect to x and simplifying we have
(34)
and(35)
Applying Newton’s methods to the system of nonlinear Equations (34) and (35) in Maple Software, we noticed that the given system of equations has no solution in .
(vi) On the face , reduces to given by
Partial derivative of with respect to c and then with respect to x, we have
(36)
and(37)
In Maple Software, we used Newton’s techniques to solve the system of nonlinear Equations (36) and (37) and observed that the above system of equations has no solution in . Thus has no optimal solution in .
3. On the Edges of Cuboid:
(i) On the edge and , then becomes
Clearly,
Putting gives the critical point at which obtains its maximum. Hence
(ii) On the edge and then takes the form
It follows that
As in we see that is decreasing over Thus has its maxima at Therefore . Thus
(iii) On the edge and then yields
(iv) On the edges and , it is noted that is free of therefore
Then
Putting we obtain the critical point at which attains its maximum. Therefore 28,952.5898. Thus
(v) On the edge and then reduces to
(vi) On the edge then becomes
is independent of x and y; therefore(vii) On the edge and then yields
Then
Since in it follows that is decreasing over Thus has its maxima at Therefore 43,200. Hence
(viii) On the edge and then takes the form
Then
The equation gives the critical point at which obtains its maximum. Therefore 23,795.60968. Hence
From above cases we conclude that
Using (31), it is clear that
The bound is sharp with the extremal function given by
□
5. Zalcman Functional
In 1960, Lawrence Zalcman proposed the following conjecture based on a coefficient for the functions belonging to the class .
Equality will be obtained when taking the Köebe function or its rotations. The particular case of the familiar Fekete–Szegö inequality will be achieved when we put . For more contributions on this particular topic, see the work [49,50].
Let f belong to and be of the form (1). Then
The inequality is sharp.
From (26) and (28), we obtain
It follows that
(38)
Let , ,, . It can be found that , and
From Lemma 3, we have
The inequality is sharp and is achieved by
□
If f belongs to and has the form (1). Then
This result is sharp.
Setting (26)–(28) with , we obtain
(39)
Using in (10)–(12), some basic calculations show that
Putting the above expressions in (39), we obtain
It can be noted that
whereBy taking and utilizing the fact we obtain
(40)
where withObviously, it can be seen that
andSince for , we obtain that for , and thus it follows that
Therefore, we have
It is not hard to calculate that
Partial derivative of with respect to c and then with respect to we have
(41)
and(42)
A numerical calculation, using Maple Software, shows that the system of Equations (41) and (42) has no solution in
For then takes the form
Then
Since in it follows that is decreasing over Thus has its maxima at Therefore Thus
For , it is easy to calculate that
andPutting , we obtain the critical point at which obtains its maximum. Therefore Thus
For then becomes
For , then reduces to
Then
Since in it is clear that is decreasing over Thus has its maxima at Hence
Thus from the above cases, we conclude that
From (31), we know that
The bound can be achieved with the extremal function given by
□
6. Sharp Bounds on the Third Hankel Determinant for Functions
Next, we will improve the bound of third Hankel determinant for which was obtained by Shi et al. in [43].
If and has the series expansion (1), then
This result is sharp.
Let From the definition, there exists a Schwarz function such that
Assuming that . Then it can be written in terms of the Schwarz function as
(43)
or equivalently,(44)
From (1), we obtain
(45)
By simplification and using the series expansion of (44), we obtain
(46)
Comparing like powers of z, , and in (45) and (46), we obtain
(47)
(48)
(49)
(50)
The third Hankel determinant can be written as
Let . It follows that
(51)
Using in (10)–(12), for some we obtain
Setting the above expressions in (51), we obtain
Thus, we see
whereTaking and utilizing the fact we obtain
(52)
where withNow, we have to maximize in the closed cuboid For this, we have to discuss the maximum values of in the interior of of its 6 faces and on its 12 edges.
1. Interior points of cuboid:
Let . Differentiating partially with respect to we obtain
Plugging we find
If is a critical point inside then , and this is only achievable if
(53)
and(54)
For the existence of critical points, we must now find solutions that meet both inequalities (53) and (54).
Let As in , it can be seen that is decreasing over Hence . It is not hard to verify that the inequality (53) cannot hold true in this situation for Thus, there is no such critical point of existing in
Suppose that there is a critical point of Q existing in the interior of cuboid . Clearly it must satisfy that . From the above discussion, it can also be known that and . Now we will prove that 1,280,000. For , by invoking and it is not hard to observe that
Therefore, we have
Obviously, it can be seen that
andSince for , we obtain that for , and thus it follows that
Therefore, we have
It is easy to calculate that attains its extremal value 1,126,373 at . Thus, we have
Hence 1,280,000. This implies that Q is less than 1,280,000 at all the critical points in the interior of . Therefore, Q has no optimal solution in the interior of .
2. Interior of all the six faces of cuboid:
(i) On the face , reduces to
Now differentiating partially with respect to , we obtain
Thus has no critical point in the interval
(ii) On the face , yields
(iii) On the face , becomes
Differentiating partially with respect to y
Taking and solving, we obtain
For the provided range of , would be a member of if with Also the derivative of partially with respect to c is
(55)
By substituting the value of y in (55), plugging and simplifying, we obtain
(56)
A calculation gives the solution of (56) in the interval that is Thus has no optimal solution in the interval
(iv) On the face , takes the form
Then
Taking and solving, we obtain Therefore 999,971.4325. Thus we have
(v) On the face , yields
Now, differentiating partially with respect to then with respect to x and simplifying, we have
(57)
and(58)
Applying Newton’s methods to the system of nonlinear Equations (57) and (58) in Maple Software, we found that the given system of equations has no solution in
(vi) On the face , reduces to
Partial derivative of with respect to c and then with respect to we have
(59)
and(60)
As mentioned in the above case, we conclude that for the face the system of Equations (59) and (60) has no solution in Thus has no optimal solution in
3. On the Edges of Cuboid:
(i) On the edge and , then becomes
It is clear that
Putting gives the critical point at which obtains its maximum. Therefore 352,004.0398. Hence
(ii) On the edge and then takes the form
Then
As in it is noted that is decreasing over Thus has its maxima at Therefore 1,280,000. Hence
(iii) On the edge and then yields
(iv) For and , as is free of it follows that
Then
Putting we obtain the critical point at which maximizes, therefore 999,971.435. Thus
(v) On the edge and then reduces to
(vi) On the edge then becomes
(vii) On the edge and then yields
Then
The equation gives the critical point at which obtains its maximum. Therefore 715,541.7526. Hence
(viii) On the edge and then takes the form
Then
Noting that in , is decreasing over Thus has its maxima at Therefore 1,280,000. Hence
From the above cases, we conclude that
Using (52) we see that
Equality is determined by the extremal function given by
□
Let f belong to with the form (1). Then
This inequality is the best one.
From (48) and (50), we obtain
After simplification, we have
(61)
Let and . It can be easily verified that and
An application of Lemma 3 leads to
The equality is obtained by
□
If f belongs to and has the expansion (1), then
This result is sharp.
Putting (48)–(50) with , we obtain
(62)
Let in (10), (11) and (12). Now using Lemma 1, we obtain
Putting the above expressions in (62), we obtain
In view of we obtain that
whereUtilizing and also observing the fact we obtain
(63)
where withNow we have to maximize in the closed cuboid For this, we have to discuss the maximum values of in the interior of in the interior of its 6 faces and on its 12 edges.
In the following, we will prove that the maximum value of is 1,280,000 in the closed cuboid . To prove this, we first discuss the maximum values of in the interior of 6 faces and 12 edges of .
It is not hard to note that , and for all . A simple calculation shows that
It is clear that
and thusFor , it follows that
For , it is noted that
Therefore, we have
Then
Putting and solving, we obtain Hence, we obtain that
Now we only need to prove that does not exceed 1,280,000 in the inside of . By observing that
we easily find that there are no critical points of S in from the proof of Theorem 8.Suppose that there is a critical point of S existing in the interior of cuboid . It is clear that . Moreover, it can be seen that and . For , by invoking and it is not hard to observe that
andTherefore, we have
It is easily to be seen that
Thus, we obtain
It is easy to calculate that attains its extremal value 1,156,314 at . Thus, we have
Hence 1,280,000. This implies that S is less than 1,280,000 at all the critical points in the interior of . Therefore, S has no optimal solution in the interior of .
From the above discussion, we conclude that
In virtue of (63), we can write
Equality is achieved by an extremal function
□
From (6), one can easily deduce the following functions
(64)
and(65)
Both these functions belong to the class Comparing coefficients of like powers of (64) and (1), we have
Then, it follows that
and
Similarly, using (65), we easily obtain that
7. Concluding Remarks and Observations
Due to the great importance of coefficients in the field of function theory, Pommerenke [16,17] proposed the topic of studying the Hankel determinant with entry of coefficients. In the current article, we considered two subfamilies of starlike and bounded turning functions, denoted by and , respectively. These families of univalent functions were connected by a three-leaf-shaped domain with the quantities and being subordinated to . For functions belonging to these classes, we investigated various intriguing problems containing initial coefficients. Among these problems, the sharp bounds of the Hankel determinant are extremely difficult to investigate, and we determined the sharp estimate of this determinant for functions belonging to both classes.
In proving our main results, finding the upper bounds of the Hankel determinant for functions belonging to or was transformed into a maximum value problem of a function with three variables in the domain of a cuboid. Based on an analysis of all the possibilities that the maxima might occur, we were able to determine the sharp upper bounds for these families. Numerical analysis was applied since some of the computations are quite complicated. Clearly, this approach may be used to calculate bounds for functions belonging to various subfamilies of univalent functions. However, in most cases, it is not so lucky to obtain such sharp results.
Furthermore, the application of the familiar quantum or fundamental (or q-) calculus, as (for example) in similar recent publications [51,52,53,54], might be a promising route for additional research based on our current findings.
This manuscript’s work has been contributed equally by all writers. All authors have read and agreed to the published version of the manuscript.
No data were used to support this study.
The authors state that they have no conflict of interest.
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Abstract
The theory of univalent functions has shown strong significance in the field of mathematics. It is such a vast and fully applied topic that its applications exist in nearly every field of applied sciences such as nonlinear integrable system theory, fluid dynamics, modern mathematical physics, the theory of partial differential equations, engineering, and electronics. In our present investigation, two subfamilies of starlike and bounded turning functions associated with a three-leaf-shaped domain were considered. These classes are denoted by
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1 Faculty of Physical and Numerical Sciences, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan;
2 Faculty of Computing and Information Technology, King Abdulaziz University, P.O. Box 411, Jeddah 21911, Saudi Arabia;