A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities

Mays S.Abdul Ameer; Abdul Rahman S.Juma; Hassan Hussien Ebrahim

doi:10.12688/f1000research.172490.1

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Research Article

A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities

[version 1; peer review: awaiting peer review]

Mays S.Abdul Ameer ¹, Abdul Rahman S.Juma², Hassan Hussien Ebrahim³

PUBLISHED 06 Jan 2026

Author details Author details

¹ Mathematics, Tikrit University, Tikrit, Saladin Governorate, Iraq
² Mathematics, University of Anbar, Ramadi, Al Anbar Governorate, Iraq
³ Mathematics, Tikrit University, Tikrit, Saladin Governorate, Iraq

Mays S.Abdul Ameer
Roles: Writing – Original Draft Preparation

Abdul Rahman S.Juma
Roles: Writing – Review & Editing

Hassan Hussien Ebrahim
Roles: Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS AWAITING PEER REVIEW

This article is included in the Fallujah Multidisciplinary Science and Innovation gateway.

Abstract

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.

Keywords

Analytic function, Univalent functions, Diﬀerential operator, Subordination, Bazilevic type, Fekete-Szego inequalities.

Corresponding author: Mays S.Abdul Ameer

Competing interests: No competing interests were disclosed.

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

Copyright: © 2026 S.Abdul Ameer M et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: S.Abdul Ameer M, S.Juma AR and Hussien Ebrahim H. A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities [version 1; peer review: awaiting peer review]. F1000Research 2026, 15:10 (https://doi.org/10.12688/f1000research.172490.1) First published: 06 Jan 2026, 15:10 (https://doi.org/10.12688/f1000research.172490.1) Latest published: 06 Jan 2026, 15:10 (https://doi.org/10.12688/f1000research.172490.1)

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

(1)

f (z) = z + \sum_{n = 2}^{\infty} 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 + \sum_{n = 1}^{\infty} c_{n} z^{n} .

Since $w (z) = \bar{z}$ is an analytic function on $\partial D,$ we define $w (z)$ as the Schwarz function defined on the domain's border $\partial D$ in $ℂ$ .¹^,²

The class $A$ has the following relationship with the class of Schwarz functions $w$ see Ref. 3.

(2)

h \in A \leftrightarrow h = \frac{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.

(3)

f ≺ g or f (z) ≺ g (z), (z \in 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 $f (0) = g (0) and f (Ω) \subset g (Ω)$ (see Ref. 4).

If a function $f (z)$ meets the following criteria, it is referred to as univalent starlike in $Ω$ :

Re {\frac{z \overset{´}{f (z)}}{f (z)}} > 0, (z \in Ω) .

$S^{*}$ is the name given to the old class of functions (see Ref. 3). We can write from Equation (1)

(4)

{(f (z))}^{β} = {(z + \sum_{n = 2}^{\infty} a_{n} z^{n})}^{β},

If $β$ is a real number greater than zero, we can obtain

(5)

\begin{matrix} {(f (z))}^{β} = {(z + a_{2} z^{2} + a_{3} z^{3} + a_{4} z^{4} + \dots)}^{β} . \\ {(f (z))}^{β} = z^{β} {(1 + a_{2} z + a_{3} z^{2} + a_{4} z^{3} + \dots)}^{β} . \end{matrix}

By applying Equation (5) binomial expansion, we get

\begin{array}{l} {(f (z))}^{β} = z^{β} {[1 + β (a_{2} z + a_{3} z^{2} + a_{4} z^{3} + \dots) + \frac{β (β - 1)}{2!} {(a_{2} z + a_{3} z^{2} + a_{4} z^{3} + \dots)}^{2} + \dots]}^{β} \\ {(f (z))}^{β} = z^{β} {(1 + β (a_{2} z + a_{3} z^{2} + a_{4} z^{3} + \dots))}^{β} . \end{array}

Therefore

{(f (z))}^{β} = z^{β} + β a_{2} z^{β + 1} + β a_{3} z^{β + 2} + β a_{4} z^{β + 3} + \dots .

The following represents the class of analytic functions $D_{β} :$

(6)

{(f (z))}^{β} = z^{β} + \sum_{n = 2}^{\infty} 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:

(7)

\begin{array}{l} H_{ξ, μ}^{m, λ} ({(f (z))}^{β}) = (f {(z)}^{β}) . \\ H_{1} = \frac{(λ - ξ) z}{μ + λ} H + (\frac{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} + \sum_{n = 2}^{\infty} a_{n} (β) z^{β + n - 1}] \\ H_{ξ, μ}^{1, λ} ({(f (z))}^{β}) = {(\frac{1 + (1 - β) (λ - ξ)}{μ + λ})}^{1} z^{β} + \sum_{n = 2}^{\infty} {(\frac{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})) = {(\frac{1 + (1 - β) (λ - ξ)}{μ + λ})}^{2} z^{β} + \sum_{n = 2}^{\infty} {(\frac{1 + (λ - ξ) (β + n - 2)}{μ + λ})}^{2} a_{n} (β) z^{β + n - 1} . \end{array}

In general, we have

(8)

\begin{matrix} H_{ξ, μ}^{m, λ} ({(f (z))}^{β}) = H_{1} (H_{ξ, μ}^{m - 1, λ} ({(f (z))}^{β})) \\ = {(\frac{1 + (1 - β) (λ - ξ)}{μ + λ})}^{m} z^{β} + \sum_{n = 2}^{\infty} {(\frac{1 + (λ - ξ) (β + n - 2)}{μ + λ})}^{m} a_{n} (β) z^{β + n - 1} . \end{matrix}

where

(μ > 0, ξ, λ \geq 0, m \in N_{0} = N \cup {0}; β > 0, z \in 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:

Definition 1.

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

(9)

Re {\frac{H_{ξ, μ}^{m, λ} f {(z)}^{β}}{{(\frac{1 + (1 - β) (λ - ξ)}{μ + λ})}^{m} z^{β}}} > γ | \frac{H_{ξ, μ}^{m, λ} f {(z)}^{β}}{{(\frac{1 + (1 - β) (λ - ξ)}{μ + λ})}^{m} z^{β}} - 1 | + α .

Where $(μ > 0, ξ, λ, γ \geq 0, m \in N_{0} = N \cup {0}; β > 0, z \in Ω)$ (see Ref. 6).

The above condition is equivalent to

(10)

\frac{H_{ξ, μ}^{m, λ} f {(z)}^{β}}{{(\frac{1 + (1 - β) (λ - ξ)}{μ + λ})}^{m} z^{β}} ≺ γ (z),

where

γ (z) = \frac{1 + αz}{1 - δαz}

be

univalent function

with

γ (0) = 1

and

\overset{´}{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

(11)

| f ́ (z) {(\frac{f (z)}{z})}^{β - 1} | < 1 .

If $ξ = 0$ and $m = λ = μ = 1, L_{0, 1}^{1, 1} (δ, α, γ) (f {(z)}^{β})$

(12)

\overset{´}{f} (z) {(\frac{f (z)}{z})}^{β - 1} ≺ γ (z) .

If $ξ = 0, m = λ = β = 1$ and placing

\begin{matrix} γ (z) = \frac{1 + z}{1 - z}, \\ \overset{´}{f} (z) ≺ \frac{1 + z}{1 - z} . \end{matrix}

Bazilevic was generalized to provide the class described in (9) (see Ref. 7). Where he presented this job and studied it as follows:

(13)

f (z) = {\frac{ξ}{1 + ε^{2}} \int_{0}^{z} (h (v) - iε) v^{- (1 + \frac{iξε}{1 + ε^{2}})} g {(v)}^{\frac{ξ}{1 + ε^{2}}} dv}^{\frac{1 + iε}{ξ}},

where the function

h (v)

belongs to

A

and

g (v) \in S^{*}

with

β > 0 .

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

Lemma 2.

Let $h (z) = 1 + c_{1} z + c_{2} z^{2} + c_{3} z^{3} + \dots,$ $z \in Ω$ belongs to the class $A$ and $μ \in C$ , then

$| c_{2} - μ {c_{1}}^{2} | \leq 2 max {1, | 2 μ - 1 |} .$ The following functions can produce crisp results.⁸

h (z) = \frac{1 + z^{2}}{1 - z^{2}} and h (z) = \frac{1 + z}{1 - z}, z \in Ω .

Lemma 3.

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} + \dots .

Then

| c_{0} \leq 1 | and | c_{k} | \leq 1 - {| c_{0} |}^{2} for k > 0 .

Lemma 4.

If $h (z) = 1 + c_{1} z + c_{2} z^{2} + c_{3} z^{3} + \dots, z \in Ω$ belongs to the class $A$ then

| c_{2} - γ {c_{1}}^{2} | \leq {\begin{cases} - 4 γ + 2 & if γ \leq 0 \\ 2 & if 0 \leq γ \leq 1 \\ 4 γ - 2 & if γ \geq 0 . \end{cases}

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

One of its rotations, or $0 \leq γ \leq 1$ . 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} \leq 2 0 < γ \leq \frac{1}{2},

and

| c_{2} - γ {c_{1}}^{2} | + (1 - γ) {| c_{1} |}^{2} \leq 2 \frac{1}{2} \leq 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

Theorem 1.

Let ${(f (z))}^{β} \in D_{β}$ which is given in Equation (6). If ${(f (z))}^{β}$ belongs to the class $L_{ξ, μ}^{m, λ} (δ, α, γ) .$ Then

\begin{matrix} | a_{2} (β) | \leq \frac{α (1 + δ)}{2 α {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}, \\ | a_{3} (β) | \leq \frac{α (1 + δ)}{2 α {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m} [1 - \frac{(1 - δα) A_{1} + α (1 + δ) A_{3}}{2 A_{1}}]} \end{matrix}

and

\begin{matrix} | a_{4} (β) | \leq \frac{α (1 + δ)}{2 β {(\frac{1 + (λ - ξ) (- 3 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} \\ [1 - \frac{(β - 1) A_{3} + 2 (1 - δα) A_{2}}{2 A_{2}} + \frac{3 (β - 1) (1 - δα) A_{1} A_{4} + 3 α (β - 1) (1 + δ) A_{3} A_{4}}{12 A_{1} A_{2}} - \frac{α (β - 2) (1 + δ) A_{3} A_{4}}{12 A_{1} A_{2}} - \frac{3}{12} (1 - 2 δα)] \end{matrix}

where

\begin{array}{l} A_{1} = 2 β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{2 m}, \\ A_{2} = β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{m} {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}, \\ \begin{matrix} A_{3} = (β - 1) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}, \\ A_{4} = α (1 + δ) {(\frac{1 + (λ - ξ) (- 3 - β)}{1 + (λ - ξ) (1 - β)})}^{m} . \end{matrix} \end{array}

Proof:

If ${(f (z))}^{β} \in L_{ξ, μ}^{m, λ} (δ, α, γ),$ 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

(14)

\frac{H_{ξ, μ}^{m, λ} f {(z)}^{β}}{{(\frac{1 + (1 - β) (λ - ξ)}{μ + λ})}^{m} z^{β}} ≺ γ (w (z)),

where

(15)

γ (z) = \frac{1 + αz}{1 - δαz} = 1 + α (1 + δ) z + δ α^{2} (1 + δ) z^{2} + δ^{2} α^{3} (1 + δ) z^{3} + \dots .

The definition of the function $h (z)$ is:

h (z) = \frac{1 + w (z)}{1 - w (z)} = 1 + c_{1} h + c_{2} h^{2} + c_{3} h^{3} + \dots .

As may be seen from Equation (14), $h \in A$ and

(16)

w (z) = \frac{1 - w (z)}{1 + w (z)} = \frac{1}{2} c_{1} z + \frac{1}{2} (c_{2} - \frac{1}{2} {c_{1}}^{2}) z^{2} + \frac{1}{2} (c_{3} - c_{1} c_{2} + \frac{1}{4} {c_{1}}^{3}) z^{3} + \dots .

Considering Equations (14), (15), and (16), we get at

(17)

\begin{array}{l} \frac{H_{ξ, μ}^{m, λ} f {(z)}^{β}}{{(\frac{1 + (1 - β) (λ - ξ)}{μ + λ})}^{m} z^{β}} ≺ γ (w (z)) \\ = γ (z) (\frac{h (z) - 1}{h (z) + 1}) . \\ = γ (\frac{1}{2} c_{1} z + \frac{1}{2} (c_{2} - \frac{1}{2} {c_{1}}^{2}) z^{2} + \frac{1}{2} (c_{3} - c_{1} c_{2} + \frac{1}{4} {c_{1}}^{3}) z^{3} + \dots), \\ = 1 + \frac{1}{2} α (1 + δ) c_{1} z + [\frac{1}{2} α (1 + δ) (c_{2} - \frac{1}{2} {c_{1}}^{2}) + \frac{1}{4} δ α^{2} (1 + δ) {c_{1}}^{2}] z^{2} + \\ + [\frac{1}{2} α (1 + δ) ((c_{3} - c_{1} c_{2} + \frac{1}{4} {c_{1}}^{3}) + \frac{1}{2} δ α^{2} (1 + δ) (c_{2} - \frac{1}{2} {c_{1}}^{2}) c_{1} + \frac{1}{8} δ^{2} α^{2} (1 + δ) {c_{1}}^{2}] \dots \end{array}

By use of the series expansion of

1 + \sum_{n = 2}^{\infty} {(\frac{1 + (λ - ξ) (β + n - 1)}{1 + ((λ - ξ) (1 - β)})}^{m} a_{n} (β) z^{n - 1},

we have

(18)

\begin{matrix} \frac{H_{ξ, μ}^{m, λ} f {(z)}^{β}}{{(\frac{1 + (1 - β) (λ - ξ)}{μ + λ})}^{m} z^{β}} = 1 + β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{m} a_{2} (β) z + [β a_{3} (β) \frac{β (β - 1)}{2!} {a_{2}}^{2} (β)] {(\frac{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) + \frac{β (β - 1) (β - 2)}{3!} {a_{2}}^{3} (β)] {(\frac{1 + (λ - ξ) (- 3 - β)}{1 + (λ - ξ) (1 - β)})}^{m} z^{3} + \dots . \end{matrix}

Based on Equations (17) and (18), we compare the coefficients of $z, z^{2}$ , and $z^{3}$ in Equation (14) to obtain

(19)

a_{2} (β) = \frac{α (1 + δ) c_{1}}{2 α {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{m}},

(20)

a_{3} (β) \leq \frac{α (1 + δ)}{2 α {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m} [1 -]} [c_{2} - \frac{(1 - δα) A_{1} + α (1 + δ) A_{3}}{2 A_{1}} {c_{1}}^{2}],

and

(21)

a_{4} (β) = \frac{α (1 + δ)}{2 β {(\frac{1 + (λ - ξ) (- 3 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} [c_{3} - \frac{(β - 1) A_{3} + 2 (1 - δα) A_{2}}{2 A_{2}} c_{1} c_{2} + (\frac{3 (β - 1) (1 - δα) A_{1} A_{4} + 3 α (β - 1) (1 + δ) A_{3} A_{4}}{12 A_{1} A_{2}} - \frac{α (β - 2) (1 + δ) A_{3} A_{4}}{12 A_{1} A_{2}} - \frac{3}{12 (1 - 2 δα)}) {c_{1}}^{3}] .

Given that $h (z)$ has an analytic function and is confined to $Ω$ ,

(22)

| c_{0} | \leq 1 and | c_{k} | \leq 1 - {| c_{0} |}^{2} for k > 0 .

By using Equation (22), we get

| a_{2} (β) | \leq \frac{α (1 + δ)}{2 α {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}, | a_{3} (β) | \leq \frac{α (1 + δ)}{2 α {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} [1 - \frac{(1 - δα) A_{1} + α (1 + δ) A_{3}}{2 A_{1}}] | a_{4} (β) | \leq \frac{α (1 + δ)}{2 β {(\frac{1 + (λ - ξ) (- 3 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} [1 - \frac{(β - 1) A_{3} + 2 (1 - δα) A_{2}}{2 A_{2}} + \frac{3 (β - 1) (1 - δα) A_{1} A_{4} + 3 α (β - 1) (1 + δ) A_{3} A_{4}}{12 A_{1} A_{2}} - \frac{α (β - 2) (1 + δ) A_{3} A_{4}}{12 A_{1} A_{2}} - \frac{3}{12 (1 - 2 δα)}] .

The proof is complete.

Using Theorem 1 and $γ (z) = \frac{1 + z}{1 - z}$ , we obtain the following.

Corollary 2.

Let ${(f (z))}^{β} \in D_{β}$ 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) | \leq \frac{1}{α {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{m}},

Proof:

Based on Equations (19) and (20), we have

| a_{3} (β) | \leq \frac{1}{α {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} - \frac{A_{1}}{α {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m} A_{3}}

and

| a_{4} (β) | \leq \frac{1}{β {(\frac{1 + (λ - ξ) (- 3 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} . [1 - \frac{2 (β - 1) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{2 A_{2}} + \frac{4 (2 β - 1) {(\frac{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,

Corollary 3.

Let $f (z) \in D$ which is given in Equation (6). If $f (z)$ belongs to the class $L_{ξ, μ}^{0, λ} (1, 1, γ) .$

Then

$\begin{array}{l} | a_{2} (β) | \leq 1 \\ | a_{3} (β) | \leq 1, \end{array}$

and

$| a_{4} (β) | \leq 1 .$

Theorem 4.

Let $γ (z) = 1 + α (1 + δ) z + δ α^{2} (1 + δ) z^{2} + \dots,$ where $γ (z) \in D$ and $\overset{´}{γ} (z) > 0 .$ If ${(f (z))}^{β}$ which is given in Equation (6) belongs to the class $L_{ξ, μ}^{m, λ} (δ, α, γ)$ and $φ \in ℕ,$ then

\begin{matrix} | a_{3} (β) - φ {a_{2}}^{2} (β) | \\ \leq \frac{α (1 + δ)}{β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} max {1, | (δα + δ) (\frac{[2 λ + β - 1] {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{2 β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}) |} . \\ \begin{matrix} a_{3} (β) - φ {a_{2}}^{2} (β) = \frac{α (1 + δ) c_{2}}{2 β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} - \frac{[2 βα (1 + δ) (1 - δα) {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{2 m} + (β - 1) β^{2} {(1 + δ)}^{2} {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}]}{8 β^{2} {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{2 m} {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} {c_{1}}^{2} \\ - φ \frac{β^{2} {(1 + δ)}^{2} {c_{1}}^{2}}{4 α^{2} {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{2 m}} . \end{matrix} \end{matrix}

Therefore

| a_{3} (β) - φ {a_{2}}^{2} (β) | = \frac{α (1 + δ) c_{2}}{2 β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} [c_{2} - v {c_{1}}^{2}],

where

v = \frac{1}{2} [1 - δβ + β (1 + δ) (\frac{(2 φ + β - 1) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{2 β {(\frac{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) | \leq \frac{α (1 + δ)}{β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} max {1, | δα + α (1 + δ) (\frac{[2 λ + β - 1] {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{2 β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}) |} .

The outcome for ${(f (z))}^{β}$ , which is provided as follows:

\frac{H_{ξ, μ}^{m, λ} f {(z)}^{β}}{{(\frac{1 + (1 - β) (λ - ξ)}{μ + λ})}^{m} z^{β}} ≺ γ (z^{2}),

or

\frac{H_{ξ, μ}^{m, λ} f {(z)}^{β}}{{(\frac{1 + (1 - β) (λ - ξ)}{μ + λ})}^{m} z^{β}} ≺ γ (z) .

Theroofsomplete.

The results of Lemma 4 are as follows:

Theorem 5.

Let $γ (z) = 1 + α (1 + δ) z + δ α^{2} (1 + δ) z^{2} + \dots, 0 \leq δ \leq 1$ and $0 < β \leq 1 .$ If

${(f (z))}^{β}$ given by Equation (6) $belongs to the class$ $L_{ξ, μ}^{m, λ} (δ, α, γ)$ , then $τ \in ℂ$

\begin{array}{l} | a_{3} (β) - φ {a_{2}}^{2} (β) | \\ \leq {\begin{cases} \frac{α (1 + δ)}{β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} [δα - α (1 + δ) (\frac{[2 τ + β - 1] {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{2 β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{m}})] & if τ \leq τ_{1} \\ \frac{α (1 + δ)}{β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}, & if τ_{1} \leq τ \leq τ_{2} \\ \frac{α (1 + δ)}{β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} [α (1 + δ) (\frac{[2 τ + β - 1] {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{2 β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}) - δα], & if τ \geq τ_{2}, \end{cases} \end{array}

Where

\begin{matrix} τ_{1} = \frac{2 α (δα - 1) {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{2 m} - α (1 + δ) (β - 1) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{2 α (1 + δ) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}, \\ τ_{2} = \frac{2 α (δα + 1) {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{2 m} - α (1 + δ) (β - 1) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{2 α (1 + δ) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} . \end{matrix}

Proof.

Since ${(f (z))}^{β} \in L_{ξ, μ}^{m, λ} (δ, α, γ)$ 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) | = \frac{α (1 + δ)}{2 β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} [c_{2} - t {c_{1}}^{2}],

where

t = \frac{1}{2} [1 - δ + α (1 + δ) (\frac{[2 τ + β - 1] {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{2 β {(\frac{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) | \leq \frac{α (1 + δ)}{β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} [δα - α (1 + δ) (\frac{[2 τ + β - 1] {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{2 β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{m}})] .

When $0 \leq t \leq 1,$ then $τ_{1} \leq τ \leq τ_{2}$ , and using Lemma 4, yields

| a_{3} (β) - φ {a_{2}}^{2} (β) | \leq \frac{α (1 + δ)}{β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} .

Assuming $t \geq 1,$ then $τ_{2} \leq τ .$ Additionally, using Lemma 4, we get

| a_{3} (β) - φ {a_{2}}^{2} (β) | \leq \frac{α (1 + δ)}{β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} [α (1 + δ) (\frac{[2 τ + β - 1] {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{2 β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}) - δα]

Using $γ (z) = \frac{1 + z}{1 - z}$ and Theorem 5, we arrive at the following conclusion:

Corollary 6.

Let $γ (z) = 1 + 2 z + 2 z^{2} + \dots . if {(f (z))}^{β}$ 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) | \leq {\begin{cases} \frac{2}{β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} [1 - (\frac{[2 τ + β - 1] {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{2 β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{m}})] & if τ \leq τ_{1} \\ \frac{2}{β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}, & if τ_{1} \leq τ \leq τ_{2} \\ \frac{2}{β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} [(\frac{[2 τ + β - 1] {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{2 β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}) - 1], & if τ \geq τ_{2}, \end{cases}

where

\begin{matrix} τ_{1} = \frac{- (1 - β)}{2 (1 + δ)}, \\ τ_{2} = \frac{4 β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{2 m} - 2 (1 - β) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{4 {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} . \end{matrix}

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

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

Corollary 7.

If $γ (z) = 1 + 2 z + 2 z^{2} + \dots . if {(f (z))}^{β}$ 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) | \leq {\begin{cases} 2 - 4 τ & if τ \leq τ_{1}, \\ 2 & if τ_{1} \leq τ \leq \\ 4 τ - 2 & if τ \geq τ_{2,} \end{cases} τ_{2},

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

Corollary 8.

Let $γ (z) = 1 + α (1 + δ) z + δ α^{2} (1 + δ) z^{2} + \dots, 0 \leq δ \leq 1$ and $0 < β \leq 1 . If {(f (z))}^{β}$ given by Equation (6) belongs to the class $L_{ξ, μ}^{m, λ} (δ, α, γ),$ and $τ_{1} \leq τ_{3} \leq τ_{2} .$ Let

\begin{matrix} τ_{3} = \frac{1}{2 α (1 + δ) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} \\ (2 βδαβ {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{2 m} - α (1 + δ) (1 - β) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}) . \end{matrix}

If $τ_{1} \leq τ \leq τ_{3},$ then

(23)

\leq \frac{β (1 + δ)}{2 β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} .

If $τ_{3} \leq τ \leq τ_{2}$ , then

(24)

\begin{matrix} | a_{3} (β) - τ {a_{2}}^{2} (β) | + \frac{2 β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{2 m}}{α (1 + δ) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} \\ \times [1 - δα + α (1 + δ) (\frac{[2 τ + β - 1] {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{2 β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{m}})] {| a_{2} (β) |}^{2} \\ \begin{matrix} | a_{3} (β) - τ {a_{2}}^{2} (β) | + \frac{2 β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{2 m}}{α (1 + δ) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} \\ \times [1 - δα + α (1 + δ) (\frac{[2 τ + β - 1] {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{2 β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{m}})] {| a_{2} (β) |}^{2} \\ \leq \frac{β (1 + δ)}{2 β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} \end{matrix} \end{matrix}

Proof.

For

\begin{matrix} | a_{3} (β) - τ {a_{2}}^{2} (β) | + (τ - τ_{1}) {| a_{2} (β) |}^{2} = \\ (τ - \frac{1}{2 α (1 + δ) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} [2 β (δα - 1) {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{2 m} + α (1 + δ) (β - 1) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}]) \\ \begin{matrix} \times \frac{β^{2} {(1 + δ)}^{2}}{{(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{2 m}} {| c_{1} |}^{2} + \frac{α (1 + δ)}{2 α {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} [c_{2} - t {c_{1}}^{2}] \\ = \frac{α (1 + δ)}{2 α {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} [c_{2} - t {c_{1}}^{2} (β) + t {| c_{1} (β) |}^{2}] . \end{matrix} \end{matrix}

It follows that using Lemma 4 yields

| a_{3} (β) - τ {a_{2}}^{2} (β) | + (τ - τ_{1}) {| a_{2} (β) |}^{2} \leq \frac{α (1 + δ)}{2 α {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}},

This is the inequality of Equation (24). For

\begin{matrix} | a_{3} (β) - τ {a_{2}}^{2} (β) | + (τ - τ_{1}) {| a_{2} (β) |}^{2} \\ = (\frac{2 β (αδ + 1) {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{2 m}}{2 β (1 + δ) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} - \frac{α (1 + δ) (β - 1) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}}{β (1 + δ) {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} - τ) \\ \begin{matrix} \times \frac{α^{2} {(1 + δ)}^{2}}{4 β {(\frac{1 + (λ - ξ) (- 1 - β)}{1 + (λ - ξ) (1 - β)})}^{2 m}} {| c_{1} (β) |}^{2} + \frac{α (1 + δ)}{2 β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} [c_{2} - t {c_{1}}^{2}] \\ = \frac{α (1 + δ)}{2 β {(\frac{1 + (λ - ξ) (- 2 - β)}{1 + (λ - ξ) (1 - β)})}^{m}} [| c_{2} - t {c_{1}}^{2} | + t {| c_{1} |}^{2}] . \end{matrix} \end{matrix}

Using Lemma 4, the following results

| a_{3} (β) - τ {a_{2}}^{2} (β) | + (τ - τ_{1}) {| a_{2} (β) |}^{2} \leq \frac{α (1 + δ)}{2 α {(\frac{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

1. 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.
2. Duren P: Grundlehren der mathematischen wissenchaffen. Vol. 259. . New York, NY, USA; Berlin/Heidelberg, Germany: Univalent Functions; Springer; 1983.
3. Graham I, Kohr G: Geometric function theory in one and higher dimensions. CRC Press; 2003.
4. 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.
5. 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. Publisher Full Text
6. 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. Publisher Full Text
7. Bazilevich IE: On a case of integrability in quadratures of the Loewner-Kufarev equation. Matematicheskii Sbornik. 1955; 79(3): 471–476.
8. 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.
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11. Silverman H: Univalent functions with negative coefficients. Proc. Am. Math. Soc. 1975; 51(1): 109–116. Publisher Full Text
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13. 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. Publisher Full Text
14. Ravichandran V, et al.: Fekete-Szegö inequality for certain class of analytic functions. Aust. J. Math. Anal. Appl. 2004; 1(2): 4–7.
15. Singh R: On Bazilevič functions. Proc. Am. Math. Soc. 1973; 38(2): 261–271.
16. Umar S, et al.: On a subclass related to Bazilevic functions. AIMS MATHEMATICS. 2020; 5(3): 2040–2056. Publisher Full Text
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18. Srivastava H, et al.: 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. Publisher Full Text

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¹ Mathematics, Tikrit University, Tikrit, Saladin Governorate, Iraq
² Mathematics, University of Anbar, Ramadi, Al Anbar Governorate, Iraq
³ Mathematics, Tikrit University, Tikrit, Saladin Governorate, Iraq

Mays S.Abdul Ameer
Roles: Writing – Original Draft Preparation

Abdul Rahman S.Juma
Roles: Writing – Review & Editing

Hassan Hussien Ebrahim
Roles: Writing – Review & Editing

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No competing interests were disclosed.

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

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S.Abdul Ameer M, S.Juma AR and Hussien Ebrahim H. A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities [version 1; peer review: awaiting peer review]. F1000Research 2026, 15:10 (https://doi.org/10.12688/f1000research.172490.1)

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[1] 1. 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.

[2] 2. Duren P: Grundlehren der mathematischen wissenchaffen. Vol. 259. . New York, NY, USA; Berlin/Heidelberg, Germany: Univalent Functions; Springer; 1983.

[3] 3. Graham I, Kohr G: Geometric function theory in one and higher dimensions. CRC Press; 2003.

[4] 4. 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.

[5] 5. 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. Publisher Full Text

[6] 6. 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. Publisher Full Text

[7] 7. Bazilevich IE: On a case of integrability in quadratures of the Loewner-Kufarev equation. Matematicheskii Sbornik. 1955; 79(3): 471–476.

[8] 8. 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.

[9] 9. Keogh F, Merkes E: A coefficient inequality for certain classes of analytic functions. Proc. Am. Math. Soc. 1969; 20(1): 8–12. Publisher Full Text

[10] 10. 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.

[11] 11. Silverman H: Univalent functions with negative coefficients. Proc. Am. Math. Soc. 1975; 51(1): 109–116. Publisher Full Text

[12] 12. 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.

[13] 13. 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. Publisher Full Text

[14] 14. Ravichandran V, et al.: Fekete-Szegö inequality for certain class of analytic functions. Aust. J. Math. Anal. Appl. 2004; 1(2): 4–7.

[15] 15. Singh R: On Bazilevič functions. Proc. Am. Math. Soc. 1973; 38(2): 261–271.

[16] 16. Umar S, et al.: On a subclass related to Bazilevic functions. AIMS MATHEMATICS. 2020; 5(3): 2040–2056. Publisher Full Text

[17] 17. 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.

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A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities

Abstract

Background

Methods

Results

Conclusions

Keywords

Introduction

Methods

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Definition 1.

(9)

(10)

(11)

(12)

(13)

Lemma 2.

Lemma 3.

Lemma 4.

Results and discussion

Theorem 1.

Proof:

(14)

(15)

(16)

(17)

(18)

(19)

(20)

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(22)

Corollary 2.

Proof:

Corollary 3.

Theorem 4.

Theorem 5.

Proof.

Corollary 6.

Corollary 7.

Corollary 8.

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(24)

Proof.

Conclusions

Ethical considerations

Data availability

Reporting guidelines

References

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