1. Introduction

F1000Research

2046-1402

F1000 Research Limited

London, UK

10.12688/f1000research.173313.1

Research Article

Articles

Bounds on Hankel Determinants with Fekete-Szegö Parameter for Bazilević Functions

[version 1; peer review: 1 approved, 1 approved with reservations, 1 not approved]

Khaled

Nathir

Data Curation Formal Analysis Methodology Project Administration Resources Writing – Review & Editing https://orcid.org/0009-0001-0668-9641 a 1 S.Juma

Abdul Rahman

Conceptualization Funding Acquisition Investigation Resources Supervision Writing – Original Draft Preparation Writing – Review & Editing b 1 1Mathematics, University of Anbar, Ramadi, Al Anbar Governorate, Iraq

a nat24u2004@uoanbar.edu.iq b eps.abdulrahman.juma@uoanbar.edu.iq

No competing interests were disclosed.

14 2 2026

2026

261

4 2 2026

2026

This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The author(s) is/are employees of the US Government and therefore domestic copyright protection in USA does not apply to this work. The work may be protected under the copyright laws of other jurisdictions when used in those jurisdictions.

This paper presents a systematic investigation of a newly defined comprehensive subclass of analytic and univalent functions in the open unit disk. The introduced function class constitutes a broad generalization that incorporates several significant families, particularly Yamaguchi functions and starlike functions, which themselves exhibit fundamental properties within the established Bazilevič function hierarchy. The formal construction of this class employs the Ma-Minda framework through the application of subordination principles, set-theoretic operations, and analytical formulations involving infinite series and product combinations of specific geometric expressions.

The principal contributions of this work include the derivation of sharp coefficient bounds, complete solutions for the Fekete-Szegö problem with parameters, and precise estimates for Hankel determinants containing a real parameter. The proposed class demonstrates remarkable generality, as it systematically reduces to numerous well-established function classes when its parameters are specialized to particular values within their defined domains, thereby providing a unified approach to studying these classical families.

Hankel determinant Bazileviˇc function subordination Yamaguchi function Ma-Minda function Fekete-Szegö estimate

The author(s) declared that no grants were involved in supporting this work.

1. Introduction

Geometric Function Theory (GFT), a captivating branch of complex analysis, is dedicated to studying the geometric properties of analytic functions. Its importance is also observed across numerous mathematical and physical disciplines, such as ( q − ) calculus, special functions, orthogonal polynomials, and the theory of conformal mappings.

Let A be the family of analytic functions on the open unit disk Ω = { z ∈ ℂ : | z | < 1 } , and let S be its subclass of normalized univalent functions of the form: f ( z ) = z + ∑ j = 2 ∞ a j z j , (1) satisfying the conditions f ( 0 ) = 0 and f ′ ( 0 ) = 1 .

A cornerstone of GFT is the coefficient problem, which seeks to determine the possible values of the coefficients a j and the bounds of functionals constructed from them. This problem was famously initiated by Bieberbach’s 1916 conjecture that | a j | ≤ j for all j ≥ 2 . The eventual proof of this conjecture by de Branges in 1985 ¹ underscored the profound depth of this area of research. The pursuit of sharp bounds for other coefficient functionals, such as the Fekete-Szegö functional and Hankel determinants, remains an active and central theme in GFT.

Prominent subclasses of S , including starlike, convex, close-to-convex, and Yamaguchi functions, are defined based on the geometric characteristics of their image domains. The class of Bazilević functions, introduced in, ² represents one of the largest known subclasses of S .

A powerful tool in defining these classes is the principle of subordination (denoted f ≺ F ). For two functions ( f , F ∈ A ), we say f is subordinate to F if there exists an analytic function: s ( z ) = s 1 z + s 2 z 2 + s 3 z 3 … (2) with s ( 0 ) = 0 and | s ( z ) | < 1 such that f ( z ) = F ( s ( z ) ) . If F is univalent, this is equivalent to f ( 0 ) = F ( 0 ) and f ( Ω ) ⊂ F ( Ω ) .

The Carathéodory class, consisting of functions p ( z ) = 1 + p 1 z + p 2 z 2 + … with a positive real part in Ω , plays a pivotal role in solving many problems in GFT. The Möbius function m 0 ( z ) = ( 1 + z ) / ( 1 − z ) serves as the extremal function for this class.

In a significant unification effort, Ma and Minda ³ introduced a general class using a function b ( z ) that is analytic, univalent, has a positive real part, and is characterized by a Taylor series b ( z ) = 1 + B 1 z + B 2 z 2 + … with B 1 > 0. This framework elegantly encapsulates many previously studied classes.

In this paper, we build upon these foundations to define a new and comprehensive subclass Λ ( δ , b ) of analytic-univalent functions using a combination of concepts from Bazilević, Yamaguchi, and starlike functions, subordinated to a Ma-Minda function. We then proceed to derive sharp coefficient bounds, Fekete-Szegö inequalities, and Hankel determinant estimates for functions belonging to this class, demonstrating that it generalizes several important families of functions previously studied in the literature.

1.1 Relevant subsets of analytic functions

A function Ҩ is called a Carathéodory function if it is analytic in the open unit disk Ω , has a positive real part, and is normalized such that Ҩ ( 0 ) = 1 . The class of all functions is symbolized by P . Formally, a function Ҩ belongs to P if it can be expressed as: Ҩ ( z ) = 1 + ∑ j = 1 ∞ p j z j , R e ( Ҩ ( z ) ) > 0 for all z ∈ Ω . (3)

Over the decades, this class has proven to be fundamental in solving numerous problems in GFT. A canonical and extremal function within the class P is the Möbius function: m 0 ( z ) = 1 + z 1 − z = 1 + 2 ∑ j = 1 ∞ z j ( z ∈ Ω ) , (4) there exists a fundamental relationship between Carathéodory functions Ҩ ( z ) and functions s ( z ) analytic in Ω with | s ( z ) | < 1 and s ( 0 ) = 0 . This relationship is given by the following transformation: Ҩ ( z ) = 1 + s ( z ) 1 − s ( z ) ⟺ s ( z ) = Ҩ ( z ) − 1 Ҩ ( z ) + 1 ( z ∈ Ω ) , (5) substituting the Taylor series for p ( z ) = 1 + p 1 z + p 2 z 2 + … into this relation yields the series expansion for s ( z ) : s ( z ) = 1 2 [ p 1 z + ( p 2 − p 1 2 2 ) z 2 + ( p 3 − p 1 p 2 + p 1 3 4 ) z 3 + … ] . (6)

A significant unification of several subclasses of starlike and convex functions was achieved by Ma and Minda ³ in 1994. They introduced a function b ( z ) , which is analytic and univalent with a series expansion of the form: b ( z ) = 1 + β 1 z + β 2 z 2 + β 3 z 3 + … ( β 1 > 0 , β k ∈ ℝ , z ∈ Ω ) . (7)

This function b(z) has the following key properties: ( R e ( b ( z ) ) ) > 0 , b ( 0 ) = 1 , b ′ ( 0 ) > 0 , b ( z ) maps the unit disk Ω onto a domain that is starlike with respect to 1 and symmetric about the real axis. By composing this function with s ( z ) , we obtain: b ( s ( z ) ) = 1 + β 1 s ( z ) + β 2 ( s ( z ) ) 2 + β 3 ( s ( z ) ) 3 + … = 1 + β 1 2 p 1 z + [ β 1 2 ( p 2 − p 1 2 2 ) + β 2 4 p 1 2 ] z 2 + [ β 1 2 ( p 3 − p 1 p 2 + p 1 3 4 ) + β 2 2 p 1 ( p 2 − p 1 2 2 ) + β 3 8 p 1 3 ] z 3 + … (8)

If a function f ∈ S maps the unit disk Ω onto a starlike domain, then f is classified as a starlike function. This geometric property is characterized by the analytic condition: f ∈ ST = { f : f ∈ S , R e ( z f ′ ( z ) f ( z ) ) > 0 , and z ∈ Ω } . (9)

The extremal function for the class ST is the Koebe function: k ( z ) = z ( 1 − z ) 2 .

The class ST was first presented via Alexander, ⁴ though its corresponding geometric property was actually characterized earlier via Nevanlinna in 1921. ⁵ Research into starlike functions has since expanded significantly, leading to diverse formulations and applications, as explored by authors such as Lasode and Opoola. ⁶ Later, in 1956, Yamaguchi ⁷ defined a specific subclass of S characterized by the following condition: f ∈ Υ = { f : f ∈ S , R e ( f ′ ( z ) ) > 0 , and z ∈ Ω } . (10)

Various properties of functions in the Yamaguchi class Υ -such as univalence, radii problems, partial sums, and growth, distortion, and inclusion theorems-have been demonstrated in the literature on GFT. ^{7–
9} Another significant subclass is B ( δ , γ , Ҩ , h ) , introduced by Bazilevič. ² Recognized as one of the largest subclasses of S , its functions are specified by the integral representation: f ( z ) = { ( δ + iγ ) ∫ 0 z Ҩ ( t ) h ( t ) γ t − ( 1 − iγ ) dt } 1 δ + iγ , where: δ > 0 , γ ∈ ℝ , h ∈ ST (a starlike function), Ҩ ∈ P (a Carathéodory function), and we employ the principal value for all powers. Singh ¹⁰ introduced two important subclasses: B ( δ , 0 , Ҩ , h ) = B ( 0 ) , B ( δ , 0 , Ҩ , z ) = B 1 ( 0 ) .

For: δ > 0 and z ∈ Ω , these classes are characterized by the following conditions, respectively: R e z f ′ ( z ) f ( z ) δ − 1 h ( z ) δ > 0 and R e z f ′ ( z ) f ( z ) δ − 1 z δ > 0 . (11)

Furthermore, the following inclusion relationships hold: B 1 ( δ ) ⊂ B ( δ ) ⊂ B ( δ , γ , Ҩ , h ) , B 1 ( 0 ) = B ( 0 ) = ST .

Special cases for δ = 1 are particularly noteworthy:

B ( 1 ) , characterized by R e ( z f ′ ( z ) h ( z ) ) > 0 , represents the class of close-to-convex functions.

B 1 ( 1 ) , characterized by R e ( f ′ ( z ) ) > 0 , represents the class of bounded turning functions.

2. Lemmas

Lemma 3.1.

⁵Let Ҩ ∈ CR . Then | p j | ≦ 2 ∀ j ∈ { 1 , 2 , 3 , 4 , … } .

The function (4) has a sharp inequality.

Lemma 3.2.

¹¹Let Ҩ ∈ CR . Then | p 2 − λ p 1 2 2 | ≦ { 2 ( 1 − λ ) when λ ≦ 0 2 when 0 ≦ λ ≦ 2 2 ( λ − 1 ) when λ ≦ 2 2 max { 1 , | 1 − λ | } when λ ∈ ℂ

Lemma 3.3.

¹²Let Ҩ ∈ CR . Then | u p 1 3 − v p 1 p 2 + w p 3 | ≦ 2 | u | + 2 | v − 2 u | + 2 | u − v + w | .

Lemma 3.4.

¹³Let Ҩ ∈ CR . Then for i , j ∈ { 1 , 2 , 3 , … } , | p i + j − μ p i p j | ≦ { 2 when 0 ≦ μ ≦ 1 2 | 2 μ − 1 | elsewhere .

3. Main results 3.1 A new class of analytic-univalent functions

A novel category of analytic univalent functions is introduced in this work. This class is developed by extending the foundational principles of Bazilevič functions and other established families, incorporating the theory of subordination. Definition 1.

The class Λ ( δ , b ) consists of all functions f ∈ A satisfying ( f ′ ( z ) ) δ ( z f ′ ( z ) f ( z ) ) 1 − δ ≺ b ( z ) , (12)

with 0 ≤ δ ≤ 1 and where b ( z ) is the Ma-Minda function defined in (7).

Remark 1.

This newly defined class provides a generalization of several established families. Its generality is demonstrated by the following examples: 1.

When δ = 1 and b ( z ) = m 0 ( z ) (see (4)), the subordination (12) simplifies to

f ′ ( z ) ≺ m 0 ( z ) .

Yamaguchi Functions: Choosing δ = 1 and b ( z ) = m 0 ( z ) from (4) yields the condition

f ′ ( z ) ≺ m 0 ( z ) .

This is the defining property for the class of Yamaguchi functions, referenced in (10). 3.

Starlike Functions: Taking δ = 0 with b ( z ) = m 0 ( z ) from (4) leads to z f ′ ( z ) f ( z ) ≺ m 0 ( z ) .

This subordination characterizes the family of starlike functions given in (9).

This study focuses on deriving sharp estimates for various problems within these classes. We establish upper bounds for the initial coefficients, the Fekete-Szegö functional, and specific Hankel determinants that incorporate τ > 0 .

3.2 Coefficient estimates for set <inline-formula> <mml:math display="inline"> <mml:mi>Λ</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula>

Theorem 1.

If a function f ∈ A belongs to the class Λ ( δ , b ) , then | a 2 | ≤ β 1 1 + δ (13) | a 3 | ≤ 2 β 1 + | β 2 | 2 + δ + ( 1 − δ ) β 1 2 ( 1 + δ ) 2 (14) | a 4 | ≤ β 1 + 2 | β 2 | + | β 3 | 3 + δ + ( 1 − δ ) ( 2 − δ ) β 1 3 6 ( 1 + δ ) 3 + ( 1 − δ ) β 1 2 ( 1 + δ ) ( 2 + δ ) + ( ( 1 − δ ) β 1 ) ( | β 2 | + β 1 ) ( 1 + δ ) ( 2 + δ ) + ( 1 − δ ) 2 β 1 3 2 ( 1 + δ ) 3 . (15)

And | a 5 | ≤ 5 β 1 + 3 | β 2 | + 3 | β 3 | + | β 4 | ( 4 + δ ) + ( 1 − δ ) ( 2 − δ ) 2 ( β 1 2 ( 1 + δ ) 2 ) ( 2 β 1 + | β 2 | ( 2 + δ ) + ( 1 − δ ) β 1 2 2 ( 1 + δ ) 2 ) + ( 1 − δ ) ( 2 − δ ) ( 3 − δ ) β 1 4 24 ( 1 + δ ) 4 + ( 1 − δ ) β 1 ( 1 + δ ) ( β 1 + 2 | β 2 | + | β 3 | 3 + δ + ( 1 − δ ) ( 2 − δ ) β 1 3 6 ( 1 + δ ) 3 + ( 1 − δ ) β 1 2 ( 1 + δ ) ( 2 + δ ) + ( ( 1 − δ ) β 1 ) ( | β 2 | + β 1 ) ( 1 + δ ) ( 2 + δ ) + ( 1 − δ ) 2 β 1 3 2 ( 1 + δ ) 3 ) + ( 1 − δ ) 2 ( 2 β 1 + | β 2 | 2 + δ + ( 1 − δ ) β 1 2 ( 1 + δ ) 2 ) 2 . (16)

Proof.

Let f ∈ A be a member of Λ ( δ , b ) . By the definition of subordination, there exists a Schwarz function such ( f ′ ( z ) ) ( z f ′ ( z ) f ( z ) ) 1 − δ = b ( s ( z ) ) ( z ∈ Ω ) , (17) using the binomial theorem, the left-hand side of (17) can be expressed as the series: ( f ′ ( z ) ) δ ( z f ′ ( z ) f ( z ) ) 1 − δ = 1 + ( 1 + δ ) a 2 z + 1 2 ( 2 + δ ) [ 2 a 3 − ( 1 − δ ) a 2 2 ] z 2 + ( 3 + δ ) 6 [ ( 1 − δ ) ( 2 − δ ) a 2 3 − 6 ( 1 − δ ) a 2 a 3 + 6 a 4 ] z 3 + … , (18) consequently, a comparison of the coefficients in (18) and (8) gives a 2 = β 1 p 1 2 ( 1 + δ ) , (19) a 3 = β 1 p 2 2 ( 2 + δ ) + [ β 2 − β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ] p 1 2 , (20) a 4 = 1 2 ( 3 + δ ) [ β 1 ( p 3 − p 1 p 2 + p 1 3 4 ) + β 2 p 1 ( p 2 − p 1 2 2 ) + β 3 4 p 1 3 ] − ( 1 − δ ) ( 2 − δ ) β 1 3 p 1 3 48 ( 1 + δ ) 3 + ( 1 − δ ) β 1 p 1 2 ( 1 + δ ) [ β 1 p 2 2 ( 2 + δ ) + ( β 2 − β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ) p 1 2 ] . (21)

And a 5 = β 1 2 ( 4 + δ ) ( p 4 − p 1 p 3 − p 2 2 2 + 3 p 1 2 p 2 4 − p 1 4 8 ) + β 2 p 1 2 ( 4 + δ ) ( p 3 − p 1 p 2 + p 1 3 4 ) + β 2 4 ( 4 + δ ) ( p 2 − p 1 2 2 ) 2 + 3 β 3 p 1 2 8 ( 4 + δ ) ( p 2 − p 1 2 2 ) + β 4 p 1 4 16 ( 4 + δ ) − ( 1 − δ ) ( 2 − δ ) 2 ( β 1 p 1 2 ( 1 + δ ) ) 2 ( β 1 p 2 2 ( 2 + δ ) + [ β 2 − β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ] p 1 2 ) + ( 1 − δ ) ( 2 − δ ) ( 3 − δ ) 24 ( β 1 p 1 2 ( 1 + δ ) ) 4 + ( 1 − δ ) ( β 1 p 1 2 ( 1 + δ ) ) ( 1 2 ( 3 + δ ) [ β 1 ( p 3 − p 1 p 2 + p 1 3 4 ) + β 2 p 1 ( p 2 − p 1 2 2 ) + β 3 4 p 1 3 ] − ( 1 − δ ) ( 2 − δ ) β 1 3 p 1 3 48 ( 1 + δ ) 3 + ( 1 − δ ) β 1 p 1 2 ( 1 + δ ) [ β 1 p 2 2 ( 2 + δ ) + ( β 2 − β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ) p 1 2 ] ) + ( 1 − δ ) 2 ( β 1 p 2 2 ( 2 + δ ) + [ β 2 − β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ] p 1 2 ) 2 . (22)

Apparently, (19) gives | a 2 | = β 1 2 ( 1 + δ ) | p 1 | .

Utilizing Lemma 1, we obtain the result given in (13). Additionally, from (20) it follows that a 3 ≤ β 1 | p 2 | 2 ( 2 + δ ) + [ | β 2 | + β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ] | p 1 | 2 .

The result in (14) follows from an application of Lemma 2 (for the case λ = 1 ) and Lemma 1. Proceeding from (21), we obtain: a 4 ≤ 1 2 ( 3 + δ ) [ β 1 | p 3 − p 1 p 2 + p 1 3 4 | + | β 2 | | p 1 | | p 2 − p 1 2 2 | + | β 3 | 4 | p 1 | 3 ] + ( 1 − δ ) ( 2 − δ ) β 1 3 | p 1 | 3 48 ( 1 + δ ) 3 + ( 1 − δ ) β 1 | p 1 | 2 ( 1 + δ ) [ β 1 | p 2 | 2 ( 2 + δ ) + ( | β 2 | + β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ) | p 1 | 2 ] .

Applying Lemma 3 with parameters u = 1 4 , v = w = 1 , followed by Lemma 2 with λ = 1 , and finally Lemma 1, yields the result stated in (15). Concluding this argument, Equation (22) provides: a 5 ≤ β 1 2 ( 4 + δ ) ( | p 4 − p 1 p 3 | − | p 2 | 2 2 + 3 4 | p 1 | 2 | p 2 − 1 3 p 1 2 2 | ) + | β 2 | | p 1 | 2 ( 4 + δ ) | p 3 − p 1 p 2 + p 1 3 4 | + | β 2 | 4 ( 4 + δ ) | p 2 − p 1 2 2 | 2 + 3 | β 3 | | p 1 | 2 8 ( 4 + δ ) | p 2 − p 1 2 2 | + | β 4 | | p 1 | 4 16 ( 4 + δ ) + ( 1 − δ ) ( 2 − δ ) 2 ( β 1 2 | p 1 | 2 4 ( 1 + δ ) 2 ) ( β 1 | p 2 | 2 ( 2 + δ ) + [ | β 2 | + β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ] | p 1 | 2 ) + ( 1 − δ ) ( 2 − δ ) ( 3 − δ ) 24 ( β 1 4 | p 1 | 4 16 ( 1 + δ ) 4 ) + ( 1 − δ ) ( β 1 | p 1 | 2 ( 1 + δ ) ) ( 1 2 ( 3 + δ ) [ β 1 | p 3 − p 1 p 2 + p 1 3 4 | + | β 2 | | p 1 | | p 2 − p 1 2 2 | + | β 3 | 4 | p 1 | 3 ] + ( 1 − δ ) ( 2 − δ ) β 1 3 | p 1 | 3 48 ( 1 + δ ) 3 + ( 1 − δ ) β 1 | p 1 | 2 ( 1 + δ ) [ β 1 | p 2 | 2 ( 2 + δ ) + ( | β 2 | + β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ) | p 1 | 2 ] ) + ( 1 − δ ) 2 ( β 1 | p 2 | 2 ( 2 + δ ) + [ | β 2 | + β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ] | p 1 | 2 ) 2 .

3.3 Fekete-Szegö estimates for the class <inline-formula> <mml:math display="inline"> <mml:mi>Λ</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula>

A central problem in the study of coefficient estimates for functions f in the class S is the analysis of the Fekete-Szegö functional, defined as: F S ( τ , f ) = | a 3 − τ a 2 2 | . (23)

In Geometric Function Theory (GFT), this functional, named for mathematicians Michael Fekete and Gábor Szegö, is a pivotal tool. Its historical significance stems from its role in refuting the Littlewood-Paley conjecture. Consequently, it has been the subject of extensive research for numerous subclasses of S , as documented in references like. ^{14–
22} Theorem 2.

Let f ∈ Λ ( δ , b ) . Then F S ( τ , f ) ≤ { β 1 2 + δ ( 4 ( 1 − δ ) ( 2 + δ ) β 1 δ ( 1 + δ ) 2 + β 2 β 1 − τ ( 2 + δ ) β 1 ( 1 + δ ) 2 ) when τ ≤ φ 1 , β 1 2 + δ when φ 1 ≤ τ ≤ φ 2 , − β 1 2 + δ ( 4 ( 1 − δ ) ( 2 + δ ) β 1 δ ( 1 + δ ) 2 + β 2 β 1 − τ ( 2 + δ ) β 1 ( 1 + δ ) 2 ) when τ ≥ φ 2 , β 1 2 + δ max { 1 , φ 3 } when τ ∈ C , where φ 1 = 4 ( 1 − δ ) δ + β 2 ( 1 + δ ) 2 β 1 2 ( 2 + δ ) − ( 1 + δ ) 2 β 1 ( 2 + δ ) , φ 2 = 4 ( 1 − δ ) δ + β 2 ( 1 + δ ) 2 β 1 2 ( 2 + δ ) + ( 1 + δ ) 2 β 1 ( 2 + δ ) .

And φ 3 = | 1 − λ | = | 4 ( 1 − δ ) ( 2 + δ ) β 1 δ ( 1 + δ ) 2 + β 2 β 1 − τ ( 2 + δ ) β 1 ( 1 + δ ) 2 | . (24)

Proof.

Substituting Equations (19) and (20) into the functional defined in (23) yields a 3 − τ a 2 2 = β 1 p 2 2 ( 2 + δ ) + [ β 2 − β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ] p 1 2 − τ ( β 1 p 1 2 ( 1 + δ ) ) 2 = β 1 2 ( 2 + δ ) ( p 2 − [ τ ( 2 + δ ) β 1 ( 1 + δ ) 2 − 4 ( 1 − δ ) ( 2 + δ ) β 1 δ ( 1 + δ ) 2 − β 2 β 1 + 1 ] p 1 2 2 ) .

So that F S ( τ , f ) = | a 3 − τ a 2 2 | = | β 1 2 ( 2 + δ ) ( p 2 − [ τ ( 2 + δ ) β 1 ( 1 + δ ) 2 − 4 ( 1 − δ ) ( 2 + δ ) β 1 δ ( 1 + δ ) 2 − β 2 β 1 + 1 ] p 1 2 2 ) | . (25)

If we set λ = τ ( 2 + δ ) β 1 ( 1 + δ ) 2 − 4 ( 1 − δ ) ( 2 + δ ) β 1 δ ( 1 + δ ) 2 − β 2 β 1 + 1 .

An application of Lemma 2 to the expression in (25) leads to | p 2 − λ p 1 2 2 | ≤ 2 ( 1 − λ ) = 2 ( 4 ( 1 − δ ) ( 2 + δ ) β 1 δ ( 1 + δ ) 2 + β 2 β 1 − τ ( 2 + δ ) β 1 ( 1 + δ ) 2 ) . (26)

So that when λ ≤ 0 , we have τ ≤ 4 ( 1 − δ ) δ + β 2 ( 1 + δ ) 2 β 1 2 ( 2 + δ ) − ( 1 + δ ) 2 β 1 ( 2 + δ ) . (27)

Secondly, | p 2 − λ p 1 2 2 | ≤ 2 . (28)

So that when 0 ≤ λ ≤ 2 , we have 4 ( 1 − δ ) δ + β 2 ( 1 + δ ) 2 β 1 2 ( 2 + δ ) − ( 1 + δ ) 2 β 1 ( 2 + δ ) ≤ τ ≤ 4 ( 1 − δ ) δ + β 2 ( 1 + δ ) 2 β 1 2 ( 2 + δ ) + ( 1 + δ ) 2 β 1 ( 2 + δ ) . (29)

Thirdly, | p 2 − λ p 1 2 2 | ≤ 2 ( λ − 1 ) = − 2 ( 4 ( 1 − δ ) ( 2 + δ ) β 1 δ ( 1 + δ ) 2 + β 2 β 1 − τ ( 2 + δ ) β 1 ( 1 + δ ) 2 ) . (30)

So that when λ ≥ 2 , we have τ ≥ 4 ( 1 − δ ) δ + β 2 ( 1 + δ ) 2 β 1 2 ( 2 + δ ) + ( 1 + δ ) 2 β 1 ( 2 + δ ) . (31)

Finally, if λ ∈ ℂ , then 1 − λ = 4 ( 1 − δ ) ( 2 + δ ) β 1 δ ( 1 + δ ) 2 + β 2 β 1 − τ ( 2 + δ ) β 1 ( 1 + δ ) 2 . (32)

Thus, the form of φ 3 in (24) is verified. Substituting Equations (26) through (32) into (25) completes the proof of the Theorem.

3.4 Hankel determinant estimates for the class <inline-formula> <mml:math display="inline"> <mml:mi>Λ</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> </inline-formula>

The Hankel matrix, characterized by constant entries along each ascending skew-diagonal, was first introduced by the German mathematician Hermann Hankel (1839–1873) in the mid-nineteenth century. Hankel’s initial research applied this matrix to the analysis of number sequences and their determinants. Since then, its utility has expanded significantly, with applications now encompassing factorial fractions, ²³ orthogonal polynomials, ²⁴ power series with integer coefficients, ²⁵ and the asymptotic properties of the determinants themselves. ²⁶ Within Geometric Function Theory (GFT), Pommerenke ²⁷ defined the Hankel determinant as follows: H i , j ( f ) = [ a j a j + 1 a j + 1 … … … a j + i − 1 … ⋮ ⋮ ⋮ ⋮ a j + i − 1 … … a j + 2 ( i − 1 ) ] . (33)

Here, i , j ≥ 1 , and the entries a j represent the coefficients of functions f ∈ S . Pommerenke originally applied these Hankel determinants to analyze the singularities of complex functions.

Building upon Pommerenke’s foundation, ²⁷ in a generalization of the Hankel determinant, Babalola ²⁸ introduced a Fekete-Szegö parameter τ > 0 into the structure of H i , j ( f ) , which led to the definition of the following determinants: H i , j τ ( f ) = [ a j a j + 1 a j + 1 … … … τa j + i − 1 … ⋮ ⋮ ⋮ ⋮ a j + i − 1 … … a j + 2 ( i − 1 ) ] . (34)

Note that (34) simplifies to | H 2 , 1 τ ( f ) | = | a 3 − τ a 2 2 | . (35) | H 2 , 2 τ ( f ) | = | a 2 a 4 − τ a 3 2 | . (36)

And | H 3 , 1 τ ( f ) | ≤ | a 3 | | a 2 a 4 − τ a 3 2 | + | a 4 | | a 2 a 3 − τ a 4 | + | a 5 | | a 3 − τ a 2 2 | . (37) | H 3 , 1 τ ( f ) | ≤ | a 3 | | H 2 , 2 τ ( f ) | + | a 4 | | L 2 τ ( f ) | + | a 5 | | H 2 , 1 τ ( f ) | . (38)

Where | L j τ ( f ) | = | a j a j + 1 − τ a j − 2 | ( j ∈ { 2 , 3 , 4 , … } ) . (39) Remark 2.

The following observations are noted: 1.

When τ = 1 is substituted into (34), the modified determinant reduces to the original form, i.e., H i , j τ ( f ) = H i , j ( f ) as defined in (33).

The absolute value of the second-order determinant | H 2 , 1 τ ( f ) | is equivalent to the Fekete-Szegö functional F S ( τ , f ) given in (23).

A comprehensive discussion of these properties can be found in the references. ^{3,
20,
22,
28} This study aims to determine the sharp upper bounds for the Hankel determinants in (36) and (38), treating the parameter τ as a positive real value.

Theorem 3.

Let f ∈ Λ ( δ , b ) . Then | H 2 , 2 τ ( f ) | ≤ β 1 1 + δ ( β 1 + 2 | β 2 | + | β 3 | 3 + δ + ( 1 − δ ) ( 2 − δ ) β 1 3 6 ( 1 + δ ) 3 + ( 1 − δ ) β 1 2 ( 1 + δ ) ( 2 + δ ) + ( ( 1 − δ ) β 1 ) ( | β 2 | + β 1 ) ( 1 + δ ) ( 2 + δ ) + ( 1 − δ ) 2 β 1 3 2 ( 1 + δ ) 3 ) + τ ( 2 β 1 + | β 2 | 2 + δ + ( 1 − δ ) β 1 2 ( 1 + δ ) 2 ) .

Proof.

Substituting the expressions from (19), (20), and (21) into Equation (36) yields a 2 a 4 − τ a 3 2 = ( β 1 p 1 2 ( 1 + δ ) ) ( 1 2 ( 3 + δ ) [ β 1 ( p 3 − p 1 p 2 + p 1 3 4 ) + β 2 p 1 ( p 2 − p 1 2 2 ) + β 3 4 p 1 3 ] − ( 1 − δ ) ( 2 − δ ) β 1 3 p 1 3 48 ( 1 + δ ) 3 + ( 1 − δ ) β 1 p 1 2 ( 1 + δ ) [ β 1 p 2 2 ( 2 + δ ) + ( β 2 − β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ) p 1 2 ] ) − τ ( β 1 p 2 2 ( 2 + δ ) + [ β 2 − β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ] p 1 2 ) 2 .

Such that | a 2 a 4 − τ a 3 2 | ≤ ( β 1 2 ( 1 + δ ) | p 1 | ) ( 1 2 ( 3 + δ ) [ β 1 | p 3 − p 1 p 2 + p 1 3 4 | + | β 2 | | p 1 | | p 2 − p 1 2 2 | + | β 3 | 4 | p 1 | 3 ] + ( 1 − δ ) ( 2 − δ ) β 1 3 | p 1 | 3 48 ( 1 + δ ) 3 + ( 1 − δ ) β 1 | p 1 | 2 ( 1 + δ ) [ β 1 | p 2 | 2 ( 2 + δ ) + ( | β 2 | + β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ) | p 1 | 2 ] ) + τ ( β 1 | p 2 | 2 ( 2 + δ ) + [ | β 2 | + β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ] | p 1 | 2 ) 2 .

The desired result of the theorem is then obtained through the systematic application of Lemma 1, Lemma 3 (with parameters u = 1 4 , v = w = 1 ), and Lemma 2 (with λ = 1 ).

Theorem 4.

Let f ∈ Λ ( δ , b ) . Then | L 2 τ ( f ) | ≤ 2 β 1 2 + β 1 | β 2 | ( 1 + δ ) ( 2 + δ ) + ( 1 − δ ) β 1 3 2 ( 1 + δ ) 3 + τ ( β 1 + 2 | β 2 | + | β 3 | 3 + δ + ( 1 − δ ) ( 2 − δ ) β 1 3 6 ( 1 + δ ) 3 + ( 1 − δ ) β 1 2 ( 1 + δ ) ( 2 + δ ) + ( ( 1 − δ ) β 1 ) ( | β 2 | + β 1 ) ( 1 + δ ) ( 2 + δ ) + ( 1 − δ ) 2 β 1 3 2 ( 1 + δ ) 3 ) .

Proof.

By substituting Equations (19), (20), and (21) into (39), we find that a 2 a 3 − τ a 4 = ( β 1 p 1 2 ( 1 + δ ) ) ( β 1 p 2 2 ( 2 + δ ) + [ β 2 − β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ] p 1 2 ) − τ ( 1 2 ( 3 + δ ) [ β 1 ( p 3 − p 1 p 2 + p 1 3 4 ) + β 2 p 1 ( p 2 − p 1 2 2 ) + β 3 4 p 1 3 ] − ( 1 − δ ) ( 2 − δ ) β 1 3 p 1 3 48 ( 1 + δ ) 3 + ( 1 − δ ) β 1 p 1 2 ( 1 + δ ) [ β 1 p 2 2 ( 2 + δ ) + ( β 2 − β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ) p 1 2 ] ) .

Therefore, | a 2 a 3 − τ a 4 | ≤ ( β 1 2 ( 1 + δ ) | p 1 | ) ( β 1 | p 2 | 2 ( 2 + δ ) + [ | β 2 | + β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ] | p 1 | 2 ) + τ ( 1 2 ( 3 + δ ) [ β 1 | p 3 − p 1 p 2 + p 1 3 4 | + | β 2 | | p 1 | | p 2 − p 1 2 2 | + | β 3 | 4 | p 1 | 3 ] + ( 1 − δ ) ( 2 − δ ) β 1 3 | p 1 | 3 48 ( 1 + δ ) 3 + ( 1 − δ ) β 1 | p 1 | 2 ( 1 + δ ) [ β 1 | p 2 | 2 ( 2 + δ ) + ( | β 2 | + β 1 4 ( 2 + δ ) + ( 1 − δ ) β 1 2 8 ( 1 + δ ) 2 ) | p 1 | 2 ] ) .

The result of the theorem is then obtained by applying Lemma 3 (with u = 1 4 , v = w = 1 ), Lemma 2 (with λ = 1 ), and Lemma 1.

Theorem 5.

Let f ∈ Λ ( δ , b ) . Then | H 3 , 1 τ ( f ) | ≤ ( 2 β 1 + | β 2 | 2 + δ + ( 1 − δ ) β 1 2 ( 1 + δ ) 2 ) × { β 1 1 + δ ( β 1 + 2 | β 2 | + | β 3 | 3 + δ + ( 1 − δ ) ( 2 − δ ) β 1 3 6 ( 1 + δ ) 3 + ( 1 − δ ) β 1 2 ( 1 + δ ) ( 2 + δ ) + ( ( 1 − δ ) β 1 ) ( | β 2 | + β 1 ) ( 1 + δ ) ( 2 + δ ) + ( 1 − δ ) 2 β 1 3 2 ( 1 + δ ) 3 ) + τ ( 2 β 1 + | β 2 | 2 + δ + ( 1 − δ ) β 1 2 ( 1 + δ ) 2 ) } + ( β 1 + 2 | β 2 | + | β 3 | 3 + δ + ( 1 − δ ) ( 2 − δ ) β 1 3 6 ( 1 + δ ) 3 + ( 1 − δ ) β 1 2 ( 1 + δ ) ( 2 + δ ) + ( ( 1 − δ ) β 1 ) ( | β 2 | + β 1 ) ( 1 + δ ) ( 2 + δ ) + ( 1 − δ ) 2 β 1 3 2 ( 1 + δ ) 3 ) { 2 β 1 2 + β 1 | β 2 | ( 1 + δ ) ( 2 + δ ) + ( 1 − δ ) β 1 3 2 ( 1 + δ ) 3 + τ ( β 1 + 2 | β 2 | + | β 3 | 3 + δ + ( 1 − δ ) ( 2 − δ ) β 1 3 6 ( 1 + δ ) 3 + ( 1 − δ ) β 1 2 ( 1 + δ ) ( 2 + δ ) + ( ( 1 − δ ) β 1 ) ( | β 2 | + β 1 ) ( 1 + δ ) ( 2 + δ ) + ( 1 − δ ) 2 β 1 3 2 ( 1 + δ ) 3 ) } + ( 5 β 1 + 3 | β 2 | + 3 | β 3 | + | β 4 | ( 4 + δ ) + ( 1 − δ ) ( 2 − δ ) 2 ( β 1 2 ( 1 + δ ) 2 ) ( 2 β 1 + | β 2 | ( 2 + δ ) + ( 1 − δ ) β 1 2 2 ( 1 + δ ) 2 ) + ( 1 − δ ) ( 2 − δ ) ( 3 − δ ) β 1 4 24 ( 1 + δ ) 4 + ( 1 − δ ) β 1 ( 1 + δ ) ( β 1 + 2 | β 2 | + | β 3 | 3 + δ + ( 1 − δ ) ( 2 − δ ) β 1 3 6 ( 1 + δ ) 3 + ( 1 − δ ) β 1 2 ( 1 + δ ) ( 2 + δ ) + ( ( 1 − δ ) β 1 ) ( | β 2 | + β 1 ) ( 1 + δ ) ( 2 + δ ) + ( 1 − δ ) 2 β 1 3 2 ( 1 + δ ) 3 ) + ( 1 − δ ) 2 ( 2 β 1 + | β 2 | 2 + δ + ( 1 − δ ) β 1 2 ( 1 + δ ) 2 ) 2 ) { β 1 2 + δ } .

Proof.

The result is established by applying the findings of Theorems 1, 2, 3, and 4 to Equation (38), followed by the necessary algebraic simplifications.

4. Conclusion

This study investigated the properties of a new class of functions that generalizes Yamaguchi and starlike functions, both of which hold significant importance within the well-known class of Bazilevič functions. The new class, Λ ( δ , b ) , building upon the Ma-Minda function, a new class of functions is formulated based on fundamental tenets of Geometric Function Theory (GFT). Key results derived for this class include tight upper estimates on the initial coefficients, the Fekete-Szegö functional for parameters, and the second and third-order Hankel determinants involving a parameter τ > 0 .

A notable feature of the newly defined set is its generality; it reduces to several well-known and previously studied function classes when specific parameters are chosen within their declared intervals.

Ethical considerations

The study did not involve human participants or animals.

Data availability

No data are associated with this article.

Acknowledgements

The author would like to thank all the article’s referees for their valuable and insightful comments.

References 1

Branges

: A proof of the Bieberbach conjecture. Acta Mathematica. 1985;154(1-2):137–152.

Bazilevič

: On a case of integrability by quadratures of the equation of Loewner Kufarev. Matematicheskii Sbornik. 1955;37(79):471–476.

: A unified treatment of some special classes of univalent functions. Proceedings of the Conference on Complex Analysis. International Press Inc.;1994;1994.

Alexander

: Functions which map the interior of the unit circle upon simple regions. Annals of Mathematics, Second Series. 1915;17(1):12–22.

Thomas

Tuneski

Vasudevarao

: Univalent Functions: A Primer. Berlin: Walter de Gruyter Inc.;2018.

Lasode

Opoola

: Some properties of a class of generalized Janowski-type q-starlike functions associated with Opoola q-differential operator and q-differential subordination. Communications Faculty of Sciences, University of Ankara, Series A1: Mathematics, and Statistics. 2024;73(2):349–364.

Yamaguchi

: On functions satisfying Re(f(z)/z) > 0. Proc. Am. Math. Soc. 1966;17(3):588–591.

Goel

: On functions satisfying Re(f(z)/z) > α. Publ. Math. 1971;18(16):111–117.

Srivastava

Owa

: Certain classes of analytic functions with varying arguments. J. Math. Anal. Appl. 1988;136(1):217–228.

Singh

: On Bazilevič functions. Proc. Am. Math. Soc. 1973;38:261–271.

Babalola

Opoola

: On the coefficients of a certain class of analytic functions. Advances in Inequalities for Series (1–13). Dragomir

Sofo

, editors. Hauppauge, New York: Nova Science Publishers, Inc.;2008.

Arif

Raza

Tang

: Hankel determinant of order three for familiar subsets of analytic functions related with sine functions. Open Mathematics. 2019;17(1):1615–1630.

Keogh

Merkes

: A coefficient inequality for certain classes of analytic functions. Proc. Am. Math. Soc. 1969;20(1):8–12.

Ayinla

Lasode

: Some coefficient properties of a certain family of regular functions associated with lemniscate of Bernoulli and Opoola differential operator. Malaya Journal of Matematik. 2024;12(2):218–228.

Lasode

Opoola

: Some new results on a certain subclass of analytic functions associated with q-differential operator and subordination. Palestine Journal of Mathematics. 2022;12(3):115–127.

Abdul Rahman

Al-Fayadh

Vijayalakshmi

: Upper bound on the third hankel determinant for the class of univalent functions using an integral operator. Afr. Mat. 2022;33(56):103–108.

Shehab

Juma

ARS

: Third order differential subordination for analytic functions involving convolution operator. Baghdad Science JournalOpen source preview. 2022;19(3):0581–0592. 10.21123/bsj.2022.19.3.0581

Shehab

Juma

ARS

: Application of Quasi Subordination Associated with Generalized Sakaguchi Type Functions. Iraqi Journal of ScienceOpen source preview. 2021;62(12):4885–4891. 10.24996/ijs.2021.62.12.26

Shehab

Juma

ARS

: Coefficient estimates for subclasses of regular functions. Iraqi Journal of Science Open source preview. 2018;59(3 C):1710–1716. 10.24996/ijs.2018.59.3C.16

Amao

Opoola

: Upper estimates for initial coefficients and Fekete-Szegö functional of a class of bi-univalent functions defined by means of subordination and associated with Horadan polynomials. Open Journal of Mathematical Analysis. 2023;7(2):21–30.

Awoyale

Opoola

: Coefficient bounds for p-valent functions. Open Journal of Mathematical Analysis. 2023;7(1):83–90.

Ayinla

Opoola

: The Fekete Szegö functional and second Hankel determinant for a certain subclass of analytic functions. Appl. Math. 2019;10:1071–1078.

Chu

: Hankel determinants of factorial fractions. Bull. Aust. Math. Soc. 2022;105(1):46–57.

Its

Krasovsky

: Hankel determinant and orthogonal polynomial for the Gaussian weight with a jump. Contemp. Math. 2008;458:215–248.

Cantor

: Power series with integral coefficients. Bull. Am. Math. Soc. 1963;69:362–366.

Charlier

Gharakhloo

: Asymptotics of Hankel determinants with a Laguerre-type or Jacobi-type potential and Fisher-Hartwig singularities. Adv. Math. 2021;383:107672.

Pommerenke

: On the coefficients and Hankel determinants of univalent functions. Proc. Lond. Math. Soc. 1966;41(1):111–122.

Babalola

: On coefficient determinants with Fekete-Szegö parameter. Applied Mathematics E-Notes. 2013;13:92–99.

10.5256/f1000research.191117.r468727

Reviewer response for version 1

hammad

suha jumaa

1 Referee https://orcid.org/0009-0000-8836-7162 1University of Tikrit, Tikrit, Iraq

Competing interests: No competing interests were disclosed.

25 3 2026

2026

This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

recommendation

approve-with-reservations

Recommendation: Accept the application with minor revision

Manuscript title: Bounds on Hankel Determinants with Fekete–Szegő Parameter for Bazilević Functions

The author is to be highly commended for deriving explicit and well-structured bounds for the first four coefficients a2,a3,a4 and a5 which constitutes a significant and nontrivial contribution within the framework of Geometric Function Theory. Obtaining such higher-order coefficient estimates is inherently challenging, particularly for generalized subclasses defined via subordination.

Moreover, the investigation of the Fekete–Szegő inequality with a general parameter τ\tauτ reflects a deep understanding of the structural aspects of coefficient problems. The inclusion of this parameterized functional not only broadens the scope of the results but also enhances their applicability within the theory of univalent functions.

In addition, the treatment of Hankel determinants represents a particularly valuable component of the paper. The derivation of bounds for second- and third-order Hankel determinants requires careful and delicate analysis due to the intricate dependence on coefficient interactions. The results obtained in this regard demonstrate substantial analytical effort and contribute meaningfully to the advancement of the field, especially in the context of generalized subclasses associated with Ma–Minda functions.

All the arithmetic and analytical computations presented in the manuscript, including the derivation of the bounds for the coefficients a2,a3,a4,a5 and the estimation of the Hankel determinants, are mathematically correct.

The conclusions are generally clear and summarize the main findings well.

Minor corrections needed.

1-Although the manuscript is generally clear, some sentences are long and could be shortened for easier reading.

2-The two special cases listed in Remark 1 (points 1 and 2) are essentially the same. To avoid redundancy and improve clarity, it is recommended to merge them into a single example. This will make the remark more concise and easier for readers to follow.

3-References/Bibliography: The bibliography is valuable; however, it would be helpful to include a few more recent and relevant works to further strengthen the manuscript.

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Yes

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

analytic and univalent functions

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

Khaled

Nathir

Mathematics, University of Anbar, Ramadi, Al Anbar Governorate, Iraq

Competing interests: No competing interests were disclosed.

27 3 2026

Dear Prof...

Best greetings.....

I hope this message finds you well.

Subject \ comments responses about the manuscript- F1000Res173313

(Bounds on Hankel Determinants with Fekete-Szegö Parameter for Bazilević Functions)

We sincerely thank for their valuable comments and constructive suggestions, which have helped us significantly improve the quality of our manuscript. We have carefully addressed all the points raised. Below, we provide a detailed point-by-point response.

1-Although the manuscript is generally clear, some sentences are long and could be shortened for easier reading.

Response 1: We have shortened lengthy sentences throughout the manuscript to improve readability.

Response 2: We have merged these cases into one clear example.

3-References/Bibliography: The bibliography is valuable; however, it would be helpful to include a few more recent and relevant works to further strengthen the manuscript.

Response 3: We have added several recent references (e.g., from 2024–2025) to strengthen the bibliography.

Acknowledgments: Thankful to reviewers for their valuable corrections and important suggestions

10.5256/f1000research.191117.r468724

Reviewer response for version 1

Lupas

Alina Alb

1 Referee https://orcid.org/0000-0002-2855-7535 1University of Oradea, Oradea, Romania

Competing interests: No competing interests were disclosed.

20 3 2026

2026

recommendation

approve

REVIEWER’s REPORT

On the paper

Bounds on Hankel Determinants with Fekete-Szegö Parameter for Bazilević Functions

by Nathir Khaled , Abdul Rahman S.Juma

In this paper the authors establish the properties of a new subclass of functions that generalizes Yamaguchi and starlike functions: tight upper estimates on the initial coefficients, the Fekete-Szegö functional for parameters, and the second and third-order Hankel determinants involving a parameter τ >0.

The results are new, correct and detailed. The paper is original and doesn’t contradict to ethical or policy issues, the question posed by authors is new and well defined, the methods used by authors are appropriate and well described, the data are sound and well controlled, the discussion and conclusions are well balanced, the title and abstract convey the obtained results, the writing is acceptable, the paper contains good scientific results.

The paper doesn’t require a revision.

Taking the above into consideration, I recommend the paper for indexing.

18.03.2026

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Complex Analysis

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Khaled

Nathir

Mathematics, University of Anbar, Ramadi, Al Anbar Governorate, Iraq

Competing interests: No competing interests were disclosed.

27 3 2026

Dear Prof...

Best greetings.....

I hope this message finds you well.

Subject \ comments responses about the manuscript- F1000Res173313

(Bounds on Hankel Determinants with Fekete-Szegö Parameter for Bazilević Functions)

Acknowledgments: Thankful to reviewers for their valuable corrections and important suggestions

10.5256/f1000research.191117.r459194

Reviewer response for version 1

Saliu

Afis

1 Referee https://orcid.org/0000-0002-0647-3824 1University of the Gambia, Serrekuna, The Gambia

Competing interests: No competing interests were disclosed.

6 3 2026

2026

recommendation

reject

The authors introduce a Bazilevič-type family associated with the general Ma–Minda function and investigate several coefficient-related results. While the topic is potentially interesting, the manuscript requires substantial revision in terms of structure, clarity, and scholarly presentation before it can be considered for indexing.

Below are my detailed observations and recommendations:

\begin{itemize}

\item[(1)] The abstract needs to be rewritten to make it concise, precise, and clearly aligned with the main objectives and contributions of the paper. At present, it does not adequately reflect the core findings.

\item[(2)] The first paragraph of the introduction does not relate to the main results obtained in the paper.

\item[(3)] Consequently, this paragraph should either be removed or substantially revised. Moreover, it is unclear how the class $\mathcal{S}$ admits representation (1); this point requires clarification and proper justification.

\item[(4)] Overall, the introduction is not well written. There is no clear connection between the present work and existing literature, and the motivation for the study is insufficiently articulated. Parts of Section 1.1 should be systematically merged into the introduction to improve logical flow. In addition, grammatical errors in these sections must be carefully corrected.

\item[(5)] There is a misuse of notation for Carathéodory functions in equations (5) and (6). The notation should be made consistent and mathematically accurate.

\item[(6)] The manuscript lacks sufficient and relevant references to support the stated results and contextualize the contribution within the existing body of work.

\item[(7)] The properties of Ma–Minda functions listed on page 4 are incomplete. These properties should be presented in a structured and itemized form to improve clarity and readability.

\item[(8)] Throughout most parts of the manuscript, the authors assume $f\in\mathcal{S}$ instead of $f\in A$. The choice between $f\in A$ and $f\in\mathcal{S}$ significantly affects the definitions and subsequent results. For example, in the definition of $\mathcal{ST}$, the assumption $f\in\mathcal{S}$ implicitly suggests that every univalent function is starlike, which is not true in general. For instance, the function

$f(z)=z+\frac{1}{2}z^{2}$

is univalent but not starlike with respect to the origin in the open unit disk.

\item[(9)] There are numerous punctuation errors throughout the manuscript. The paper requires careful proofreading.

\item[(10)] The overall organization of the paper is weak, particularly in the introduction and Section 1.1. A clearer structural framework is necessary.

\item[(11)] In the inclusion relation following equation (11), the symbols $\mathcal{B}_{1}(\delta)$ and $\mathcal{B}(\delta)$ are used without definition.

\item[(12)] In Lemmas 3.1--3.4, the symbol $\mathcal{CR}$ appears without prior definition.

\item[(13)] Lemma 3.1 is not properly typeset and should be reformatted for clarity and correctness.

\item[(14)] The short statement preceding the main results appears disconnected from the surrounding discussion. It should either be integrated properly or removed.

\item[(15)] In Remark 1, Special Cases 1 and 2 are identical. This redundancy should be corrected.

\item[(16)] The origin of the assumption $\tau>0$ in the final paragraph before Section 3.2 is unclear and requires justification.

\item[(17)] The authors should explicitly state whether their results are sharp. If so, appropriate justification and extremal functions must be provided; otherwise, the validity and optimality of the coefficient bounds remain uncertain.

\item[(18)] Finally, the manuscript contains numerous grammatical errors that must be thoroughly corrected to meet academic standards.

\end{itemize}

\begin{center}

\section*{Recommendation}

\end{center}

In its current form, the manuscript requires major revision, particularly with respect to mathematical rigor, organization, clarity of definitions, and linguistic accuracy. A careful restructuring and substantial editorial improvement are strongly recommended.

Is the work clearly and accurately presented and does it cite the current literature?

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Are the conclusions drawn adequately supported by the results?

Partly

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Complex Analysis (Geometric function theory)

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

Khaled

Nathir

Mathematics, University of Anbar, Ramadi, Al Anbar Governorate, Iraq

Competing interests: No competing interests were disclosed.

27 3 2026

Dear Prof...

Best greetings.....

I hope this message finds you well.

Subject \ comments responses about the manuscript- F1000Res173313

(Bounds on Hankel Determinants with Fekete-Szegö Parameter for Bazilević Functions)

Response 1: We have rewritten the abstract to be more concise and focused on the main objectives, methodology, and key results. The revised abstract now clearly highlights the class definition, the derived sharp bounds, and the significance of the results.

Response 2 & 3: We have removed the irrelevant opening paragraph and clarified the representation of class S. Equation (1) now explicitly states the normalization conditions f(0)=0 and f′(0)=1, and we have added a brief justification for the series form.

Response 4: We have restructured the introduction entirely. Section 1.1 has been merged into the introduction to create a logical flow. We added a clear motivation, linked our work to existing literature, and corrected grammatical errors throughout.

Response 5: We have corrected the notation. The Carathéodory function is now consistently denoted by p(z) instead of CR, and equations (5) and (6) have been revised accordingly.

Response 6: We have added several recent and relevant references to better contextualize our work within the existing literature, particularly regarding Hankel determinants, Fekete–Szegő functionals, and Ma–Minda functions..

Response 7: We have restructured the properties of the Ma–Minda function into a clear, itemized list for improved readability.

Response 8: We have corrected this issue throughout the manuscript. The definitions now clearly use f ∈ A where appropriate, and the class S is reserved for univalent functions. The definition of starlike functions has been corrected to require f ∈ S only after the geometric condition is imposed.

Response 9: The entire manuscript has been carefully proofread, and punctuation errors have been corrected.

Response 10: We have reorganized the manuscript. The introduction now integrates all preliminary concepts, and the structure follows a logical progression: introduction → lemmas → main results (definition, coefficient bounds, Fekete–Szegő, Hankel determinants) → conclusion.

Response 11: We have added clear definitions for these symbols following equation (11).

Response 12 & 13: We have replaced CR with the standard notation P (Carathéodory class) throughout the lemmas. Lemma 1 has been reformatted for clarity.

Response 14: This statement has been integrated into the introduction or removed as appropriate.

Response 15: We have merged these two cases into a single example and corrected the remark accordingly.

Response 16: We have added a justification for τ > 0 in Section 3.4, explaining that it serves as a generalized Fekete–Szegő parameter and that the case τ = 1 reduces to the classical Hankel determinant.

Response 17: We have revised the manuscript to explicitly state sharpness where applicable and provided extremal functions (e.g., the Koebe function and Möbius transformations) for the coefficient bounds.

Response 18: A thorough language revision has been performed to meet academic standards.

Acknowledgments: Thankful to reviewers for their valuable corrections and important suggestions