F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.172490.1 Research Article Articles A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities

[version 1; peer review: 1 approved, 1 approved with reservations]

 S.Abdul Ameer Mays Writing – Original Draft Preparation https://orcid.org/0009-0002-0471-0483 a 1 S.Juma Abdul Rahman Writing – Review & Editing 2 Hussien Ebrahim Hassan Writing – Review & Editing 3 Mathematics, Tikrit University, Tikrit, Saladin Governorate, Iraq Mathematics, University of Anbar, Ramadi, Al Anbar Governorate, Iraq Mathematics, Tikrit University, Tikrit, Saladin Governorate, Iraq mays33@st.tu.edu.iq

No competing interests were disclosed.

 6 1 2026 2026 15 10 15 12 2025 Copyright: © 2026 S.Abdul Ameer M et al. 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. Background

The Fekete–Szegö inequality and coefficient bounds play a fundamental role in the study of Bazilevič-type functions, a subclass of univalent functions with significant applications in complex analysis and related mathematical fields. These functions have been extensively studied for their geometric properties and coefficient behavior, yet there remains a need to explore subclasses defined via differential operators and subordination techniques to obtain sharper and more general bounds. Understanding the coefficient structure of these functions provides insight into their analytic behavior and extends classical results in the theory of univalent functions.

 Methods

This study focuses on a specific subclass of Bazilevič-type functions and investigates its properties using differential subordination and differential operators. These mathematical tools are employed to derive explicit coefficient estimates and establish the Fekete–Szegö inequalities. The analysis involves careful application of operator techniques to obtain bounds that reflect the influence of the subclass parameters on the analytic functions under consideration.

 Results

The derived inequalities demonstrate the effectiveness of differential operators in obtaining precise coefficient constraints. The results highlight how variations in the defining parameters of the subclass influence the function behavior, providing clear and explicit bounds for the coefficients. These findings extend existing results in the literature and offer new insights into the structure of Bazilevič-type functions.

 Conclusions

Overall, this study provides a systematic approach to understanding the coefficient structure of Bazilevič-type functions. The findings establish a foundation for further theoretical research on univalent functions and offer tools that can be applied in both theoretical investigations and practical problems in complex analysis. The results contribute to the broader understanding of analytic functions and their applications in mathematical modeling.

 Analytic function Univalent functions Differential operator Subordination Bazilevic type Fekete-Szego inequalities. The author(s) declared that no grants were involved in supporting this work. Introduction

Geometric function theory is a subfield of complex analysis that studies the geometric characteristics of analytic functions. The foundation of complex analysis is academic research on univalent and multivalent function theory. find it fascinating because of its complex geometry and variety of research options. Understanding univalent functions is crucial for the complicated analysis of single and multiple variables. Univalent functions, which are analytic and one-to-one in the unit disk, form a fundamental class in complex analysis and geometric function theory, with numerous applications in mathematical modeling and theoretical investigations. 1 – 3 Among the various subclasses, Bazilevič-type functions, introduced by Bazilevich, 4 – 7 have drawn significant attention due to their rich geometric properties and connections with differential subordination. 8 These functions have been widely studied for their coefficient bounds and associated inequalities, including the classical Fekete–Szegö problem, which provides important constraints on the coefficients of analytic and multivalent functions. 9 – 12 Recent developments have focused on extending these studies to subclasses defined via differential operators and linear transformations, allowing for the derivation of sharper coefficient estimates and Fekete–Szegö inequalities. 13 , 14 Investigations on bi-Bazilevič functions, Faber polynomial coefficients, and operator-based approaches have further highlighted the effectiveness of these methods in obtaining explicit bounds and revealing structural properties of analytic functions. 15 Despite these advances, there remains a need to explore new subclasses of Bazilevič-type functions associated with specific differential or linear operators, particularly in relation to their coefficient estimates and Fekete–Szegö inequalities. 16 – 18 This study addresses this gap by examining the subclass, establishing explicit coefficient bounds, and formulating relevant Fekete–Szegö inequalities. The findings provide a systematic framework for understanding the analytic and geometric properties of Bazilevič-type functions and offer potential directions for future research in the theory of univalent functions.

 Methods

Consider the open unit disc to be Ω = { z : | z | < 1 } and let D stand for the class of analytic functions of the type f ( z ) = z + ∑ n = 2 ∞ a n z n .

The functions are normalized with f ( 0 ) = 0 and f ′ ( 0 ) = 1 , and they are analytic in the open unit disk.

In the open unit disk Ω , let E represent the subclass of all univalent functions.

A refers to the class of functions h ( z ) with a positive real fraction in Ω . As follows. h ( z ) = 1 + ∑ n = 1 ∞ c n z n .

Since w ( z ) = z ¯ is an analytic function on ∂ D , we define w ( z ) as the Schwarz function defined on the domain's border ∂ D in ℂ . 1 , 2

The class A has the following relationship with the class of Schwarz functions w see Ref. 3. h ∈ A ↔ h = 1 + w 1 − w

Consider two analytic functions in Ω , f and h, then we say f ( z ) is subordinate to g ( z ) . Write as follows. f ≺ g or f ( z ) ≺ g ( z ) , ( z ∈ Ω ) ,

If there is an analytic Schwarz function in Ω with w ( 0 ) = 0 and | w ( z ) | < 1 , then

f ( z ) = g ( w ( z ) ) . If a function g ( z ) be univalent in Ω , then f ( z ) ≺ g ( z ) i s equivalent to f ( 0 ) = g ( 0 ) and f ( Ω ) ⊂ g ( Ω ) (see Ref. 4).

If a function f ( z ) meets the following criteria, it is referred to as univalent starlike in Ω : Re { z f ( z ) ´ f ( z ) } > 0 , ( z ∈ Ω ) .

S ∗ is the name given to the old class of functions (see Ref. 3). We can write from Equation (1) ( f ( z ) ) β = ( z + ∑ n = 2 ∞ a n z n ) β ,

If β is a real number greater than zero, we can obtain ( f ( z ) ) β = ( z + a 2 z 2 + a 3 z 3 + a 4 z 4 + … ) β . ( f ( z ) ) β = z β ( 1 + a 2 z + a 3 z 2 + a 4 z 3 + ⋯ ) β .

By applying Equation (5) binomial expansion, we get ( f ( z ) ) β = z β [ 1 + β ( a 2 z + a 3 z 2 + a 4 z 3 + ⋯ ) + β ( β − 1 ) 2 ! ( a 2 z + a 3 z 2 + a 4 z 3 + ⋯ ) 2 + … ] β ( f ( z ) ) β = z β ( 1 + β ( a 2 z + a 3 z 2 + a 4 z 3 + ⋯ ) ) β .

Therefore ( f ( z ) ) β = z β + β a 2 z β + 1 + β a 3 z β + 2 + β a 4 z β + 3 + ⋯ .

The following represents the class of analytic functions D β : ( f ( z ) ) β = z β + ∑ n = 2 ∞ a n ( β ) z β + n − 1 .

G ( D β ) is the differential operator determined for a function ( f ( z ) ) β given in Equation (6) on the space of analytic functions.

H ξ , μ m , λ : G ( D β ) ⟶ G ( D β ) as follows: H ξ , μ m , λ ( ( f ( z ) ) β ) = ( f ( z ) β ) . H 1 = ( λ − ξ ) z μ + λ H + ( 1 − ( λ − ξ ) μ + λ ) H ξ , μ 1 , λ ( ( f ( z ) ) β ) = H 1 ( H ξ , μ 0 , λ ( ( f ( z ) ) β ) ) = H 1 [ z β + ∑ n = 2 ∞ a n ( β ) z β + n − 1 ] H ξ , μ 1 , λ ( ( f ( z ) ) β ) = ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) 1 z β + ∑ n = 2 ∞ ( 1 + ( n − β ) ( λ − ξ ) μ + λ ) 1 a n ( β ) z β + n − 1 H ξ , μ 2 , λ ( ( f ( z ) ) β ) = H 1 ( H ξ , μ 1 , λ ( ( f ( z ) ) β ) ) = ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) 2 z β + ∑ n = 2 ∞ ( 1 + ( λ − ξ ) ( β + n − 2 ) μ + λ ) 2 a n ( β ) z β + n − 1 .

In general, we have H ξ , μ m , λ ( ( f ( z ) ) β ) = H 1 ( H ξ , μ m − 1 , λ ( ( f ( z ) ) β ) ) = ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β + ∑ n = 2 ∞ ( 1 + ( λ − ξ ) ( β + n − 2 ) μ + λ ) m a n ( β ) z β + n − 1 . where ( μ > 0 , ξ , λ ≥ 0 , m ∈ N 0 = N ∪ { 0 } ; β > 0 , z ∈ Ω ) .

The goal of this study is to find “coefficient bounds and Fekete-Szego inequalities for the subclass L ξ , μ m , λ ( δ , α , γ ) of Bazilevic type functions”.

Now, using the differential operator, define a class of Bazilevic functions (see Ref. 5) as follows:

The function f ( z ) β which has the form Equation (5) belongs to the class L ξ , μ m , λ ( δ , α , γ ) satisfies the following conditions Re { H ξ , μ m , λ f ( z ) β ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β } > γ | H ξ , μ m , λ f ( z ) β ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β − 1 | + α .

Where ( μ > 0 , ξ , λ , γ ≥ 0 , m ∈ N 0 = N ∪ { 0 } ; β > 0 , z ∈ Ω ) (see Ref. 6).

The above condition is equivalent to H ξ , μ m , λ f ( z ) β ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β ≺ γ ( z ) , where γ ( z ) = 1 + αz 1 − δαz be univalent function with γ ( 0 ) = 1 and γ ´ ( z ) > 0 .

If ξ = δ = 0 and m = λ = μ = α = 1 , we derive the following definition of the subclass of Bazilevic univalent functions. 4 | f ́ ( z ) ( f ( z ) z ) β − 1 | < 1 .

If ξ = 0 and m = λ = μ = 1 , L 0 , 1 1 , 1 ( δ , α , γ ) ( f ( z ) β ) f ´ ( z ) ( f ( z ) z ) β − 1 ≺ γ ( z ) .

If ξ = 0 , m = λ = β = 1 and placing γ ( z ) = 1 + z 1 − z , f ´ ( z ) ≺ 1 + z 1 − z .

Bazilevic was generalized to provide the class described in (9) (see Ref. 7). Where he presented this job and studied it as follows: f ( z ) = { ξ 1 + ε 2 ∫ 0 z ( h ( v ) − iε ) v − ( 1 + iξε 1 + ε 2 ) g ( v ) ξ 1 + ε 2 dv } 1 + iε ξ , where the function h ( v ) belongs to A and g ( v ) ∈ S ∗ with β > 0 .

The following lemmas are necessary for you to discuss the main results.

Let h ( z ) = 1 + c 1 z + c 2 z 2 + c 3 z 3 + … , z ∈ Ω belongs to the class A and μ ∈ C , then

| c 2 − μ c 1 2 | ≤ 2 max { 1 , | 2 μ − 1 | } . The following functions can produce crisp results. 8 h ( z ) = 1 + z 2 1 − z 2 and h ( z ) = 1 + z 1 − z , z ∈ Ω .

Let the function h ( z ) is analytic in Ω, with | h ( z ) | < 1 and let h ( z ) = c 0 + c 1 z + c 2 z 2 + c 3 z 3 + ⋯ .

Then | c 0 ≤ 1 | and | c k | ≤ 1 − | c 0 | 2 for k > 0 .

If h ( z ) = 1 + c 1 z + c 2 z 2 + c 3 z 3 + ⋯ , z ∈ Ω belongs to the class A then | c 2 − γ c 1 2 | ≤ { − 4 γ + 2 if γ ≤ 0 2 if 0 ≤ γ ≤ 1 4 γ − 2 if γ ≥ 0 .

For γ < 0 or γ > 1 , equality holds if and only if h ( z ) equals 1 + z 1 − z or one of them is a rotation. If 0 < γ < 1 , then equality holds if and only if h ( z ) is equal to 1 + z 2 1 − z 2 or one of them is rotations. Inequality becomes equality when γ = 0 if and only if h ( z ) = ( 1 + λ 2 ) 1 + z 1 − z + ( 1 − λ 2 ) 1 + z 1 − z ,

One of its rotations, or 0 ≤ γ ≤ 1 . If and only if the function h is the reciprocal of one of the functions such that equality holds in the case of γ = 0 , equality holds for γ = 1 .

If the highest limit stated above is sharp, it can be made better when 0 < γ < 1 : | c 2 − γ c 1 2 | + γ | c 1 | 2 ≤ 2 0 < γ ≤ 1 2 , and | c 2 − γ c 1 2 | + ( 1 − γ ) | c 1 | 2 ≤ 2 1 2 ≤ γ < 1 .

Many writers have examined Bazilevic functions, including Refs. 5, 10– 13 and 14. Studied Singh in 1973, “two subclasses of the class B-the class of Bazilevic functions”. 15 In 2020, Particle Umar et al. investigated a subclass related to Bazilevic functions. 16 Juma and AL-khafaj 2019 The characteristics and properties of a family of Bazilevic (type) functions that are provided by particular linear operators. 17 Faber polynomial coefficient estimates for a subclass of analytic Bi-Bazilevic functions defined by a differential operator were introduced in Refs. 6, 18.

 Results and discussion

Let ( f ( z ) ) β ∈ D β which is given in Equation (6). If ( f ( z ) ) β belongs to the class L ξ , μ m , λ ( δ , α , γ ) . Then | a 2 ( β ) | ≤ α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , | a 3 ( β ) | ≤ α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ 1 − ( 1 − δα ) A 1 + α ( 1 + δ ) A 3 2 A 1 ] and | a 4 ( β ) | ≤ α ( 1 + δ ) 2 β ( 1 + ( λ − ξ ) ( − 3 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ 1 − ( β − 1 ) A 3 + 2 ( 1 − δα ) A 2 2 A 2 + 3 ( β − 1 ) ( 1 − δα ) A 1 A 4 + 3 α ( β − 1 ) ( 1 + δ ) A 3 A 4 12 A 1 A 2 − α ( β − 2 ) ( 1 + δ ) A 3 A 4 12 A 1 A 2 − 3 12 ( 1 − 2 δα ) ] where A 1 = 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m , A 2 = β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , A 3 = ( β − 1 ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , A 4 = α ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 3 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m .

If ( f ( z ) ) β ∈ L ξ , μ m , λ ( δ , α , γ ) , then from Equation (10) there is the Schwarz function w ( z ) . This is analytic in Ω, with w ( 0 ) = 0 and | w ( z ) | < 1 such that H ξ , μ m , λ f ( z ) β ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β ≺ γ ( w ( z ) ) , where γ ( z ) = 1 + αz 1 − δαz = 1 + α ( 1 + δ ) z + δ α 2 ( 1 + δ ) z 2 + δ 2 α 3 ( 1 + δ ) z 3 + ⋯ .

The definition of the function h ( z ) is: h ( z ) = 1 + w ( z ) 1 − w ( z ) = 1 + c 1 h + c 2 h 2 + c 3 h 3 + ⋯ .

As may be seen from Equation (14), h ∈ A and w ( z ) = 1 − w ( z ) 1 + w ( z ) = 1 2 c 1 z + 1 2 ( c 2 − 1 2 c 1 2 ) z 2 + 1 2 ( c 3 − c 1 c 2 + 1 4 c 1 3 ) z 3 + ⋯ .

Considering Equations (14), (15), and (16), we get at H ξ , μ m , λ f ( z ) β ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β ≺ γ ( w ( z ) ) = γ ( z ) ( h ( z ) − 1 h ( z ) + 1 ) . = γ ( 1 2 c 1 z + 1 2 ( c 2 − 1 2 c 1 2 ) z 2 + 1 2 ( c 3 − c 1 c 2 + 1 4 c 1 3 ) z 3 + ⋯ ) , = 1 + 1 2 α ( 1 + δ ) c 1 z + [ 1 2 α ( 1 + δ ) ( c 2 − 1 2 c 1 2 ) + 1 4 δ α 2 ( 1 + δ ) c 1 2 ] z 2 + + [ 1 2 α ( 1 + δ ) ( ( c 3 − c 1 c 2 + 1 4 c 1 3 ) + 1 2 δ α 2 ( 1 + δ ) ( c 2 − 1 2 c 1 2 ) c 1 + 1 8 δ 2 α 2 ( 1 + δ ) c 1 2 ] …

By use of the series expansion of 1 + ∑ n = 2 ∞ ( 1 + ( λ − ξ ) ( β + n − 1 ) 1 + ( ( λ − ξ ) ( 1 − β ) ) m a n ( β ) z n − 1 , we have H ξ , μ m , λ f ( z ) β ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β = 1 + β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m a 2 ( β ) z + [ β a 3 ( β ) β ( β − 1 ) 2 ! a 2 2 ( β ) ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m z 2 + [ β a 4 ( β ) + β ( β − 1 ) a 2 ( β ) a 3 ( β ) + β ( β − 1 ) ( β − 2 ) 3 ! a 2 3 ( β ) ] ( 1 + ( λ − ξ ) ( − 3 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m z 3 + … .

Based on Equations (17) and (18), we compare the coefficients of z , z 2 , and z 3 in Equation (14) to obtain a 2 ( β ) = α ( 1 + δ ) c 1 2 α ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , a 3 ( β ) ≤ α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ 1 − ] [ c 2 − ( 1 − δα ) A 1 + α ( 1 + δ ) A 3 2 A 1 c 1 2 ] , and a 4 ( β ) = α ( 1 + δ ) 2 β ( 1 + ( λ − ξ ) ( − 3 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ c 3 − ( β − 1 ) A 3 + 2 ( 1 − δα ) A 2 2 A 2 c 1 c 2 + ( 3 ( β − 1 ) ( 1 − δα ) A 1 A 4 + 3 α ( β − 1 ) ( 1 + δ ) A 3 A 4 12 A 1 A 2 − α ( β − 2 ) ( 1 + δ ) A 3 A 4 12 A 1 A 2 − 3 12 ( 1 − 2 δα ) ) c 1 3 ] .

Given that h ( z ) has an analytic function and is confined to Ω , | c 0 | ≤ 1 and | c k | ≤ 1 − | c 0 | 2 for k > 0 .

By using Equation (22), we get | a 2 ( β ) | ≤ α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , | a 3 ( β ) | ≤ α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ 1 − ( 1 − δα ) A 1 + α ( 1 + δ ) A 3 2 A 1 ] | a 4 ( β ) | ≤ α ( 1 + δ ) 2 β ( 1 + ( λ − ξ ) ( − 3 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ 1 − ( β − 1 ) A 3 + 2 ( 1 − δα ) A 2 2 A 2 + 3 ( β − 1 ) ( 1 − δα ) A 1 A 4 + 3 α ( β − 1 ) ( 1 + δ ) A 3 A 4 12 A 1 A 2 − α ( β − 2 ) ( 1 + δ ) A 3 A 4 12 A 1 A 2 − 3 12 ( 1 − 2 δα ) ] .

The proof is complete.

Using Theorem 1 and γ ( z ) = 1 + z 1 − z , we obtain the following.

Let ( f ( z ) ) β ∈ D β which is given in Equation (6). If ( f ( z ) ) β belonging to the class L ξ , μ m , λ ( 1 , 1 , γ ) . Then | a 2 ( β ) | ≤ 1 α ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ,

Based on Equations (19) and (20), we have | a 3 ( β ) | ≤ 1 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m − A 1 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m A 3

and | a 4 ( β ) | ≤ 1 β ( 1 + ( λ − ξ ) ( − 3 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m . [ 1 − 2 ( β − 1 ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 A 2 + 4 ( 2 β − 1 ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m A 3 12 A 1 A 2 ]

We obtain the result by entering α = 1 and m = 0 in Corollary 2,

Let f ( z ) ∈ D which is given in Equation (6). If f ( z ) belongs to the class L ξ , μ 0 , λ ( 1 , 1 , γ ) .

Then

| a 2 ( β ) | ≤ 1 | a 3 ( β ) | ≤ 1 ,

and

| a 4 ( β ) | ≤ 1 .

Let γ ( z ) = 1 + α ( 1 + δ ) z + δ α 2 ( 1 + δ ) z 2 + … , where γ ( z ) ∈ D and γ ´ ( z ) > 0 . If ( f ( z ) ) β which is given in Equation (6) belongs to the class L ξ , μ m , λ ( δ , α , γ ) and φ ∈ ℕ , then | a 3 ( β ) − φ a 2 2 ( β ) | ≤ α ( 1 + δ ) β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m max { 1 , | ( δα + δ ) ( [ 2 λ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) | } . a 3 ( β ) − φ a 2 2 ( β ) = α ( 1 + δ ) c 2 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m − [ 2 βα ( 1 + δ ) ( 1 − δα ) ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m + ( β − 1 ) β 2 ( 1 + δ ) 2 ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ] 8 β 2 ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m c 1 2 − φ β 2 ( 1 + δ ) 2 c 1 2 4 α 2 ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m .

Therefore | a 3 ( β ) − φ a 2 2 ( β ) | = α ( 1 + δ ) c 2 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ c 2 − v c 1 2 ] , where v = 1 2 [ 1 − δβ + β ( 1 + δ ) ( ( 2 φ + β − 1 ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m ) ] .

Using Lemma 2, we obtain | a 3 ( β ) − φ a 2 2 ( β ) | ≤ α ( 1 + δ ) β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m max { 1 , | δα + α ( 1 + δ ) ( [ 2 λ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) | } .

The outcome for ( f ( z ) ) β , which is provided as follows: H ξ , μ m , λ f ( z ) β ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β ≺ γ ( z 2 ) , or H ξ , μ m , λ f ( z ) β ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β ≺ γ ( z ) . Theroofsomplete.

The results of Lemma 4 are as follows:

Let γ ( z ) = 1 + α ( 1 + δ ) z + δ α 2 ( 1 + δ ) z 2 + ⋯ , 0 ≤ δ ≤ 1 and 0 < β ≤ 1 . If

( f ( z ) ) β given by Equation (6) belongs to the class L ξ , μ m , λ ( δ , α , γ ) , then τ ∈ ℂ | a 3 ( β ) − φ a 2 2 ( β ) | ≤ { α ( 1 + δ ) β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ δα − α ( 1 + δ ) ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) ] if τ ≤ τ 1 α ( 1 + δ ) β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , if τ 1 ≤ τ ≤ τ 2 α ( 1 + δ ) β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ α ( 1 + δ ) ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) − δα ] , if τ ≥ τ 2 ,

Where τ 1 = 2 α ( δα − 1 ) ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m − α ( 1 + δ ) ( β − 1 ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 α ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , τ 2 = 2 α ( δα + 1 ) ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m − α ( 1 + δ ) ( β − 1 ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 α ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m .

Since ( f ( z ) ) β ∈ L ξ , μ m , λ ( δ , α , γ ) and γ ( z ) given by Equation (17), then a 2 ( β ) and a 3 ( β ) are given in Theorem 1. Furthermore, | a 3 ( β ) − φ a 2 2 ( β ) | = α ( 1 + δ ) 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ c 2 − t c 1 2 ] , where t = 1 2 [ 1 − δ + α ( 1 + δ ) ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m ) ] .

Using Lemma 4, we can establish the inequality Equation (10) as follows. | a 3 ( β ) − φ a 2 2 ( β ) | ≤ α ( 1 + δ ) β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ δα − α ( 1 + δ ) ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) ] .

When 0 ≤ t ≤ 1 , then τ 1 ≤ τ ≤ τ 2 , and using Lemma 4, yields | a 3 ( β ) − φ a 2 2 ( β ) | ≤ α ( 1 + δ ) β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m .

Assuming t ≥ 1 , then τ 2 ≤ τ . Additionally, using Lemma 4, we get | a 3 ( β ) − φ a 2 2 ( β ) | ≤ α ( 1 + δ ) β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ α ( 1 + δ ) ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) − δα ]

Using γ ( z ) = 1 + z 1 − z and Theorem 5, we arrive at the following conclusion:

Let γ ( z ) = 1 + 2 z + 2 z 2 + ⋯ . if ( f ( z ) ) β which is given in Equation (6) belongs to the class L ξ , μ m , λ ( δ , α , γ ) , then for any real complex number σ | a 3 ( β ) − φ a 2 2 ( β ) | ≤ { 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ 1 − ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) ] if τ ≤ τ 1 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , if τ 1 ≤ τ ≤ τ 2 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) − 1 ] , if τ ≥ τ 2 , where τ 1 = − ( 1 − β ) 2 ( 1 + δ ) , τ 2 = 4 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m − 2 ( 1 − β ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 4 ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m .

Observe that we obtain the following Corollary for and in Corollary 6.

For β = 1 and m = 0 , Corollary 6 yields the following.

If γ ( z ) = 1 + 2 z + 2 z 2 + ⋯ . if ( f ( z ) ) β given by Equation (6) belongs to the class L ξ , μ 0 , λ ( 1 , 1 , γ ) .

Then for any real complex number σ | a 2 ( β ) − τ a 1 2 ( β ) | ≤ { 2 − 4 τ if τ ≤ τ 1 , 2 if τ 1 ≤ τ ≤ 4 τ − 2 if τ ≥ τ 2 , τ 2 ,

The result of W. Ma. D. Minda [Ref. 8, Lemma 1] is obtained when τ 1 = 0 and τ 2 = 1 .

Let γ ( z ) = 1 + α ( 1 + δ ) z + δ α 2 ( 1 + δ ) z 2 + ⋯ , 0 ≤ δ ≤ 1 and 0 < β ≤ 1 . If ( f ( z ) ) β given by Equation (6) belongs to the class L ξ , μ m , λ ( δ , α , γ ) , and τ 1 ≤ τ 3 ≤ τ 2 . Let τ 3 = 1 2 α ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ( 2 βδαβ ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m − α ( 1 + δ ) ( 1 − β ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) .

If τ 1 ≤ τ ≤ τ 3 , then ≤ β ( 1 + δ ) 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m .

If τ 3 ≤ τ ≤ τ 2 , then | a 3 ( β ) − τ a 2 2 ( β ) | + 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m α ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m × [ 1 − δα + α ( 1 + δ ) ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) ] | a 2 ( β ) | 2 | a 3 ( β ) − τ a 2 2 ( β ) | + 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m α ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m × [ 1 − δα + α ( 1 + δ ) ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) ] | a 2 ( β ) | 2 ≤ β ( 1 + δ ) 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m

For | a 3 ( β ) − τ a 2 2 ( β ) | + ( τ − τ 1 ) | a 2 ( β ) | 2 = ( τ − 1 2 α ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ 2 β ( δα − 1 ) ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m + α ( 1 + δ ) ( β − 1 ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ] ) × β 2 ( 1 + δ ) 2 ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m | c 1 | 2 + α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ c 2 − t c 1 2 ] = α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ c 2 − t c 1 2 ( β ) + t | c 1 ( β ) | 2 ] .

It follows that using Lemma 4 yields | a 3 ( β ) − τ a 2 2 ( β ) | + ( τ − τ 1 ) | a 2 ( β ) | 2 ≤ α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ,

This is the inequality of Equation (24). For | a 3 ( β ) − τ a 2 2 ( β ) | + ( τ − τ 1 ) | a 2 ( β ) | 2 = ( 2 β ( αδ + 1 ) ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m 2 β ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m − α ( 1 + δ ) ( β − 1 ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m β ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m − τ ) × α 2 ( 1 + δ ) 2 4 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m | c 1 ( β ) | 2 + α ( 1 + δ ) 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ c 2 − t c 1 2 ] = α ( 1 + δ ) 2 β   ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ | c 2 − t c 1 2 | + t | c 1 | 2 ] .

Using Lemma 4, the following results | a 3 ( β ) − τ a 2 2 ( β ) | + ( τ − τ 1 ) | a 2 ( β ) | 2 ≤ α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ,

which is the inequality of Equation (24).

Conclusions

In this study, we have investigated a subclass of Bazilevič-type functions associated with the class of univalent functions and examined its analytic properties using differential subordination and differential operators. By applying these techniques, new coefficient estimates and Fekete–Szegö inequalities were derived, providing sharper and more general results than several existing findings in the literature. The obtained results demonstrate that the parameters defining the subclass play a significant role in determining the coefficient behavior and geometric characteristics of the associated analytic functions. The study contributes to the deeper understanding of the coefficient structure of Bazilevič-type functions and extends the theoretical framework of univalent function theory. Furthermore, the approach used in this work may be applied to other subclasses of analytic or multivalent functions to establish similar inequalities and functional relationships. Future research may focus on exploring additional subclasses generated by modified differential operators or by incorporating complex parameters that enhance the geometric interpretation of these functions in applied fields such as fluid dynamics and conformal mapping.

 Ethical considerations

This study did not involve human participants or animals, and therefore, ethical approval was not required.

 Data availability

No data are associated with this article.

Reporting guidelines

This manuscript is paper in the field of Complex Analysis. Therefore, no specific reporting guidelines (such as CONSORT, PRISMA, or STROBE, which apply to empirical or experimental studies) are applicable. The paper follows standard mathematical reporting practices: all definitions, theorems, and proofs are presented with full logical rigor, and all symbols and notations are clearly defined upon first use. The structure and exposition conform to conventional mathematical standards for reproducibility and clarity.

 References Abdul Ameer MS Juma ARS Al-Saphory RA : On differential subordination of higher-order derivatives of multivalent functions. J. Phys. Conf. Ser. 2021. IOP Publishing. Duren P : Grundlehren der mathematischen wissenchaffen. Vol.259. New York, NY, USA; Berlin/Heidelberg, Germany: Univalent Functions; Springer;1983. Graham I Kohr G : Geometric function theory in one and higher dimensions. CRC Press;2003. Hamzat J Fagbemiro O : Some properties of a new subclass of Bazilevic functions defined by Catas et al differential operator. FUW Trends Food Sci. Technol. 2018;3(28):909–917. Orhan H Raducanu D : The Fekete–Szegö functional for generalized starlike and convex functions of complex order. Asian-Eur. J. Math. 2021;14(03):2150036. 10.1142/S1793557121500364 Juma ARS Abdulhussain MS Al-khafaji SN : Faber Polynomial Coefficient Estimates for Subclass of Analytic Bi-Bazilevic Functions Defined by Differential Operator. Baghdad Sci. J. 2019;16(1):33. 10.21123/bsj.2019.16.1(Suppl.).0248 Bazilevich IE : On a case of integrability in quadratures of the Loewner-Kufarev equation. Matematicheskii Sbornik. 1955;79(3):471–476. Ma W : A unified treatment of some special classes of univalent functions. in Proceedings of the Conference on Complex Analysis. Vol.1992. International Press Inc.;1992. Keogh F Merkes E : A coefficient inequality for certain classes of analytic functions. Proc. Am. Math. Soc. 1969;20(1):8–12. 10.1090/S0002-9939-1969-0232926-9 Juma A Ularu N : Fekete-Szego inequality for certain subclasses of multivalent functions associated with quasi-sub ordination. Lib. Math. 2016;36(1):45–52. Silverman H : Univalent functions with negative coefficients. Proc. Am. Math. Soc. 1975;51(1):109–116. 10.1090/S0002-9939-1975-0369678-0 Srivastava H Mishra A Das M : The fekete-szegö-problem for a subclass of close-to-convex functions. Complex Var. Elliptic Equat. 2001;44(2):145–163. Bucur R Andrei L Breaz D : Coefficient bounds and Fekete-Szegö problem for a class of analytic functions defined by using a new differential operator. Appl. Math. Sci. 2015;9(25-28):1355–1368. 10.12988/ams.2015.511 Ravichandran V : Fekete-Szegö inequality for certain class of analytic functions. Aust. J. Math. Anal. Appl. 2004;1(2):4–7. Singh R : On Bazilevič functions. Proc. Am. Math. Soc. 1973;38(2):261–271. Umar S : On a subclass related to Bazilevic functions. AIMS MATHEMATICS. 2020;5(3):2040–2056. 10.3934/math.2020135 Juma A Al-khafaji S Irmak H : Properties and characteristics of a family consisting of Bazilevic (type) functions specified by certain linear operators. Electron. J. Math. Anal. Appl. 2019;7:39–47. Srivastava H : Faber polynomial coefficient inequalities for bi-Bazilevič functions associated with the Fibonacci-number series and the square-root functions. J. Inequal. Appl. 2024;2024(1):16. 10.1186/s13660-024-03090-9 10.5256/f1000research.190223.r459388 Reviewer response for version 1 Oros Georgia Irina 1 Referee https://orcid.org/0000-0003-2902-4455 University of Oradea, Oradea, Romania

Competing interests: No competing interests were disclosed.

 10 3 2026 Copyright: © 2026 Oros GI 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

As the Abstract adequately describes, the paper investigates a specific subclass of Bazilevič-type functions, utilizing differential operators and subordination techniques to establish Fekete–Szegö inequalities and coefficient bounds. The methods used for developing the new results are classical in Geometric Function Theory, particularly differential subordination, but well used for giving complete and correct proofs for the new results. The motivation of the study is well argued in Introduction. The history of the study is given supported by well chosen and properly cited references. However, it is suggested to add a few references of recent studies from 2025, 2025 on Bazilevič-type functions. The paper is adequately structured and all the necessary information is offered in the Methods part. It is advised to highlight the known results mentioned by adding the reference in brackets, next to the definitions or lemmas. There is no reference given for Lemma 2, 3 and 4. Please indicate where they were taken from. The new outcome is established in the Results and Discussion part. The proofs are complete and correct. This is reinforced by the mention of previously obtained results that are generalized by the original outcome of this study that can be obtained by taking specific values of the parameters involved in the newly proved theorems. The corollaries that follow the new results are given for functions with particular geometric properties which makes those corollaries interesting and inspiring for applications.  

Overall, the paper is well written and presents interesting new results.

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:

Geometric Function Theory

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.

*Saleh Mays

Competing interests: No competing interests were disclosed.

 25 3 2026

We sincerely thank the reviewer for their positive and encouraging evaluation of our manuscript.

We are grateful for your confirmation that the work is clearly presented, technically sound, and supported by appropriate methodology and analysis. We also appreciate your acknowledgment of the adequacy of the data, reproducibility, and the validity of the conclusions.

Your expertise in Geometric Function Theory and your supportive assessment are highly valued.

Thank you again for your time and thoughtful review.

 10.5256/f1000research.190223.r459389 Reviewer response for version 1 Murugusundaramoorthy Gangadharan 1 Referee https://orcid.org/0000-0001-8285-6619 Vellore Institute of Technology, Vellore, India

Competing interests: No competing interests were disclosed.

 9 3 2026 Copyright: © 2026 Murugusundaramoorthy G 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

In page 4, after line 5, by applying Equation (5) binomial expansion, we get -------- to be checked and ^\beta to be removed May english corrections

Notational mistakes. Certain things are not defined like H and H_1

The linear transformation in (7) is not defined properly.

References are given properly in standard style format

Results are correct

Methodology techniques are okay.

The definition of the operator is to be given clearly.

After incorporating these points I will recommend this article for indexing.

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 GEOMETRIC FUNCTION THEORY-30C45

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.