1. Introduction

Complex analysis, a fundamental pillar of modern mathematical science, has far-reaching implications across various academic and scientific disciplines. Within its scope, Geometric Function Theory (GFT) stands out as a fascinating branch that delves into the geometric properties of analytical functions. This field has emerged as a vital tool in applied mathematics, with significant applications in engineering, electronics, fluid dynamics, nonlinear integral systems, contemporary mathematical physics, and differential equation, further highlighting the profound impact of complex analysis on our understanding of the world.

The Bieberbach conjecture, a historic problem in GFT, has been a focal point of research for over a century. Formulated by Bieberbach [1] in 1916, the conjecture pertains to the bounds of coefficients for univalent functions, which are a fundamental class of injective functions in complex analysis. The conjecture specifically states that for a function h in the class , with a Taylor-Maclaurin series expansion of (1), the coefficients are bounded by , for all a relationship that has far-reaching implications for the field. The set , first explored by Koebe in 1907, is a suclass of the larger class of analytic functions. Bieberbach’s work in 1916 marked the beginning of progress on this conjecture, with subsequent breakthroughs by Lowner [2], Garabedian and Schiffer [3], Pederson and Schiffer [4], and Pederson [5], gradually extending the proof to cover the cases , and 6. The conjecture was ultimately resolved for all in 1985 by de Branges [6], who employed hypergeometric functions to deliver a comprehensive solution to this long-standing problem in complex analysis. The application of coefficient bounds to medical imaging data can improve the precision of diagnostic modalities such as MRI and CT scans, ultimately leading to the development of more targeted and effective treatments for a variety of health conditions.

Let U denote the unit disk, defined as and let denote a class of analytic functions satisfying the normalization conditions

For each can be written as:

(1)

A function h is considered univalent in U if it has a one-to-one correspondence between U and its image, meaning that each point in the image has a unique inverse image in U. In other words, for all if then Let is the notation used to represent the set of all functions that have the property of univalence.

The class , consisting of analytic functions p that satisfy the normalization condition and have positive real part, i.e., for all t in the domain U with

(2)

As is widely recognized, functions of the form (1) have an analytic inverse in the domain , as stated by Koebe’s theorem. Specifically, if h belongs to the class , its inverse can be expanded as

(3)

Lowner [2] demonstrated that for functions h in S with inverses of this form, the coefficients A_n are bounded by the sharp estimate

(4)

The Koebe function’s inverse, is known to produce the sharpest bounds for coefficients |A_n| in (4) across all members of the class S. This has led to considerable interest in exploring the behavior of inverse coefficients for functions h in specific geometric subclasses of S, as defined in (3). Although multiple authors have offered alternative proofs of the inequality (4), Yang proof [7] stands out for its simplicity and clarity. Using the fact that h(h⁻¹(w)) = w, it is straightforward to observe from Eq 3 that

(5)

During the period from 1916 to 1985, significant research efforts were focused on establishing estimates for the coefficient bounds of different subclasses within the family of class , such as starlike (, close-to-convex (K), convex (C) and other related families of functions. The following definitions apply to these families:

Suppose u is Schwarz analytic function in U, with and for all t in U. If h(t) and g(t) are analytic in U, and for all t in U, then h is subordinate to g, written as . Furthermore, if g is univalent in U and , then .

Ma and Minda [8] replaced the function with a more general analytic function which satisfies the specific conditions , , and maps U onto univalently a region starlike with respect to 1 and symmetric with respect to the real axis. They investigated a comprehensive and general class of functions that encompasses various prominent classes as specific instances, providing a unified framework that includes various special cases:

The mathematical community recognizes the functions in the class as Ma-Minda starlike functions, a designation honoring the contributions of mathematicians Ma and Minda. This general class has given rise to a diverse array of subfamilies, which have been extensively studied in (see, [9, 10]). For instance, Kumar et al. [11] have considered generated function of Bell numbers

In recent research, Mendiratta et al. [12] investigated the exponential function

and Goel and Kumar [13] have extensively investigated the Sigmoid function

They obtained important results on its structural representation, inclusion properties, coefficient bounds, growth behavior, distortion estimates, subordination relationships, and radii constants, respectively. Furthermore, Deniz [14] addressed the sharp coefficient problem in 2021, focusing on the specific function , a generating function for generalized telephone numbers. Meanwhile, Murugusundaramoorthy et al. [15] investigated -bi-pseudo-starlike functions with respect to symmetric points associated with Telephone numbers.

Geometric function theory has long investigated the upper bound for coefficients, which offers valuable insights into function behavior. The second coefficient bound is particularly crucial, as it leads to growth and distortion theorems, and additionally, the coefficient problem connected to Hankel determinants is another significant area of study. A key mathematical object in this study is the Hankel determinant. This determinant H_q,n(h), are defined by

(6)

This concept was introduced by Pommerenke [16, 17]. Significant research, including the works of [18, 19], has focused on determining sharp bounds for the second-order Hankel determinant, notably and , within various subfamilies of class . For a comprehensive understanding, see [11–13]. The third-order Hankel determinant, H_3,1(h), poses a considerable challenge, especially in establishing sharp bounds. The determinant H_3,1(h) is

and has been extensively studied in [23–25]. Babalola [26] first investigated H_3,1(h) for the , , and families. Zaprawa [27] later extended these findings in 2017, proposing non-sharp bounds:

Research efforts continued to enhance these bounds, specifically for the class [28, 29]. Ultimately, the sharp bounds for these determinants were established for the , , and classes, as presented in [30–32]:

Further advancements by Barukab et al. [33] and Lecko et al. [34] have determined the sharp bounds for for other classes:

Those seeking a deeper understanding of third-order Hankel determinants in recently discovered subfamilies of univalent functions are encouraged to explore the works cited in [35, 36] for further details and analysis:

A class of starlike functions related with three-leaf-shaped region, was introduced by Gandhi in [43] as:

Tang et al. [44] studied the class

of symmetric points with three-leaf-shaped region. In 2021, Mustafa and Murugusundaramoorthy [45] introduced Mocanu-type bi-starlike functions associated with shell-shaped regions, deriving coefficient bounds and Hankel determinants. Separately, Murugusundaramoorthy et al. [46] explored coefficient functionals for a class of bounded turning functions connected to the three-leaf function. Inspired by aforementioned investigation, we explore the function , which transforms U into a starlike region centered at 1, with coefficients that are the Gregory coefficients. These coefficients, similar to Bernoulli numbers, are rational numbers that decrease in value ,... and are essential in various numerical analysis and number theory contexts. They were initially discovered by James Gregory in 1671 and have since been a subject of interest in mathematics. Gregory coefficients are notably among the most frequently rediscovered mathematical entities. These coefficients have been identified by various names, including reciprocal logarithmic numbers, the second kind of Bernoulli numbers, Cauchy numbers, and others. Our focus in this paper is on the generating function G_n of Gregory coefficients, as described in (see [47, 48]), which is defined as follows:

(7)

Clearly, g_n for are

In the Fig 1, we describe the image behavior of Gregory coefficients under the unit disk.

Recently, Bulut [49] research focused on applying Faber polynomial methods to analytic bi-univalent functions associated with Gregory coefficients, introducing the following class definition:

Definition 1. [49]. For A function defined in (1) is in the class if

where denote the class of bi-univalent functions, is defined by (7) and g(w) = h⁻¹(t).

Murugusundaramoorthy et al. [50] investigated three classes of bi-univalent functions related to Gregory coefficients, defining them as follows:

Definition 2. [50]. A function defined in (1) is in the class if

where denote the class of bi-univalent functions and g(w) = h⁻¹(t).

Definition 3. [50]. For A function defined in (1) is in the class if

where denote the class of bi-univalent functions and g(w) = h⁻¹(t).

Definition 4. [50]. For A function defined in (1) is in the class if

where denote the class of bi-univalent functions and g(w) = h⁻¹(t).

Firstly, they demonstrated that these classes are non-empty. Furthermore, for functions within each of these three bi-univalent function classes, Murugusundaramoorthy et al. [50] investigated the initial estimates and of the Taylor–Maclaurin coefficients and Fekete–Szego functional problems. Building on the aforementioned research, we now define a new class of symmetric starlike functions, denoted by , which are related to the symmetric points associated with Gregory coefficients. This class is characterized by the following properties:

Definition 5. Suppose that the function , as defined by the Eq 1, if

(8)

where is given in (7).

2. Set of Lemmas

Lemma 1. ([51],[52]). Suppose the function p belongs to the class , and is defined by Eq 2. Then

(9)(10)(11)(12)(13)

Lemma 2. [53]. Suppose the function p belongs to the class , and is defined by Eq 2. Then

(14)

and

(15)

for some with and .

Lemma 3. Suppose the function , and is defined by Eq 2, then

(16)

and if and then

(17)

For complex numbers we have

(18)

See [52–56] for the inequality (16), (17) and (18).

Lemma 4. [57]. Suppose the function p belongs to the class , and is defined by Eq 2, 0 < T₂ < 1, 0 < Q₁ < 1 and

(19)

Then

(20)

Lemma 5. [58] Let . Also, for , let . If then

Furthermore, if RM<0, then

where

3. Main results

In the following result, we establish initial bounds for the function

Theorem 1. Assume the function h, defined by (1), is in the class , then

The estimates provided are sharp and achieved by the functions given in (29)–(32), respectively.

Proof: Given that h is a member of , and using the definition of the Schwarz function, we obtain

(21)

The function p is defined as

then . This implies that

(22)

It is evident that p is analytic in the region U, satisfying , and

By using (22) and , we get

and

(23)

It follows by (21), (22) and (23) that

(24)(25)(26)(27)

Using the inequality (9) of Lemma 1, on a₂, we get

Rearrange (25), we have

(28)

Using the inequality (10) of Lemma 1, on (28), we have

Rearranging (26) and (27) it gives

where

It gives us 0 < F < 1, and Therefore by using the Lemma 3, we have

Again rearrange (27) we have

where

Now we have

and

Hence (19) of the Lemma 4 is satisfy, therefore using the inequality (20) of Lemma 4, we have

The bounds, and are sharp for the following extremal functions:

(29)(30)(31)(32)

Therefore, the proof is now complete.

Theorem 2. Suppose that . Then, the following sharp estimates hold:

Theorem 2 is sharp for the function defined in (30).

Proof: Using (24) and (25), we obtain

Using the inequality (18) of Lemma 3, we have

Therefore, the proof is now complete.

Corollary 1. Suppose . Then

The estimate is sharp for the function provided in Eq 30.

Theorem 3. Suppose . Then

(33)

Theorem 3 is sharp for the function provided in Eq 29.

Proof: Using (24), (25) and (26), we get

where

It gives us 0 < F < 1, E < F and . Therefore by using the inequality (17) of Lemma 3, we have

Therefore, the proof is now complete.

Theorem 4. Suppose that . Then

The estimate is sharp for the function provided in Eq 30.

Proof: From (24), (25) and (26), we have

Applying the Lemma 2 and let , we can write

Since and if s = 0, then . Therefore

If then

(34)

Suppose that . Then, we have

where

Consequently, RM > 0. Furthermore, it is readily apparent that

Thus, we get

Let l = s², and . Then

In that case, we have

(35)

Therefore, the proof is now complete.

Theorem 5. Assume that . Then

Proof: Since by (6), we have

By using Theorem 1, Theorem 1, Theorem 3 and Theorem 4, we have

Therefore, the proof is now complete.

Theorem 6. Assume the function h, defined by (1), belongs to and has the power series representation ..., then

(36)

The estimates provided are sharp and achieved by the functions given in (29)-(31), respectively.

Proof: If and . Substituting Eq 24 into equation (5), we get

Using the inequality (9) of Lemma 1, on A₂, we have

Now for A₃, substituting Eqs (24), (25) in (5), we have

Thus

By the inequality (10) of Lemma 1, we have

Now for A₄, so from (5), we have

where

It gives us 0 < F < 1, E < F and . Therefore by using the inequality (17) of Lemma 3, we have

Therefore, the proof is now complete.

Logarithmic function

The logarithmic coefficients L_n of a function , which belongs to the class , are defined by the following formula:

(37)

The logarithmic coefficients have far-reaching implications in the study of univalent functions, and their impact is evident in numerous estimates. De Branges [59] seminal work in 1985 demonstrated that

(38)

and the equality holds if and only if h has the specific form , where is a real number.

This inequality is a cornerstone of univalent function theory, encompassing the celebrated Bieberbach-Robertson-Milin conjectures on Taylor coefficients. Andreev and Duren [60] notably employed logarithmic coefficients to establish Brennan’s conjecture for conformal mappings. The study of logarithmic coefficients has since flourished, with significant contributions from, Alimohammadi et al. [61], Deng [62], Roth [63], Ye [64], and Girela [65]. Their work has substantially expanded our knowledge of logarithmic coefficients in various subclasses of holomorphic univalent functions. According to the definition, the logarithmic coefficients for a function h in are easily calculated as:

(39)(40)(41)(42)

Theorem 7. Assume that . Then

(43)

Proof: Let . Then, using the Eqs (24), (25), (26) and (27), in (39), (40), (41) and (42), we get

(44)(45)(46)

and

(47)

For L₁, using the inequality (9) of Lemma 1 , on (44), we obtain

For L₂, rearrange (45), thus

Using the inequality (10) of Lemma 1, we have

For L₃, rearrange (46) as:

where

It gives us 0<F<1, E<F and F(2F-1)<E<F. Therefore by using the inequality (17) of Lemma 3, we have

For L₄, rearrange (47) as:

where

It follows that

and

Hence (19) of the Lemma 4 is satisfy, therefore using the inequality (20) of Lemma 4, we have

Therefore, the proof is now complete.

4. Conclusion

This research has established a new class, , of symmetric starlike functions, which are connected to the generating function of Gregory coefficients through a subordination relationship. Our investigation has led to the discovery of various coefficient inequalities, including sharp coefficient bounds, Fekete-Szego problems and an upper bound for the third-order Hankel determinant, and inverse inequalities. Finally, we have also derived sharp estimates for the logarithmic and inverse coefficients of functions in the class of symmetric starlike functions. Future research directions include further exploration of the class to determine Toeplitz and higher-order Hankel determinants, Extreme point theorem, Partial sums results, Necessary and sufficient conditions, Convex combination, Closure theorem, Growth and distortion bounds, Radii of close-to-starlikeness and starlikeness.