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СпутникиРНSpaceXРоскосмосNASABackground 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.
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.
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+∑n=2∞anzn.The functions are normalized with f(0)=0andf′(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∞cnzn.
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.
(2)h∈A↔h=1+w1−wConsider two analytic functions in Ω , f and h, then we say f(z) is subordinate to g(z) . Write as follows.
(3)f≺gorf(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)andf(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{zf(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)
(4)(f(z))β=(z+∑n=2∞anzn)β,If β is a real number greater than zero, we can obtain
(5)(f(z))β=(z+a2z2+a3z3+a4z4+…)β.(f(z))β=zβ(1+a2z+a3z2+a4z3+⋯)β.By applying Equation (5) binomial expansion, we get
(f(z))β=zβ[1+β(a2z+a3z2+a4z3+⋯)+β(β−1)2!(a2z+a3z2+a4z3+⋯)2+…]
(f(z))β=zβ(1+β(a2z+a3z2+a4z3+⋯).
(f(z))β=zβ+βa2z(β+1)+βa3z(β+2)+βa4z(β+3)+⋯.
The following represents the class of analytic functions Dβ:
(6)(f(z))β=zβ+∑n=2∞an(β)zβ+n−1.G(Dβ) is the differential operator determined for a function (f(z))β given in Equation (6) on the space of analytic functions.
Let f∈G(Dβ), where (f(z))β=zβ+∑n=2∞an(β)zβ+n−1.
Hξ,μm,λ:G(Dβ)⟶G(Dβ)
Hξ,μ0,λ((f(z))β)=(f(z))β,
Hξ,μm,λ((f(z))β=H1((Hξ,μ)(m−1,λ)((f(z))β),m∈ℕ,
where the auxiliary operator H1 is defined by
H1((f(z))β)=1+(1−β)(λ−ξ)μ+λzβ+∑n=2∞1+(λ−ξ)(β+n−2)μ+λ2an(β)zβ+n−1.
Hξ,μ1,λ((f(z))β)=H1(Hξ,μ0,λ((f(z))β))=H1[zβ+∑n=2∞an(β)zβ+n−1]
(7)Hξ,μ1,λ((f(z))β)=(1+(1−β)(λ−ξ)μ+λ)1zβ+∑n=2∞(1+(n−β)(λ−ξ)μ+λ)1an(β)zβ+n−1Hξ,μ2,λ((f(z))β)=H1(Hξ,μ1,λ((f(z))β))=(1+(1−β)(λ−ξ)μ+λ)2zβ+∑n=2∞(1+(λ−ξ)(β+n−2)μ+λ)2an(β)zβ+n−1.
(8)Hξ,μm,λ((f(z))β)=(1+(1−β)(λ−ξ)μ+λ)mzβ+∑n=2∞(1+(λ−ξ)(β+n−2)μ+λ)man(β)zβ+n−1.where (μ>0,ξ,λ≥0,m∈N0=N∪{0};β>0,z∈Ω).
The goal of this study is to find “coefficient bounds and Fekete-Szego inequalities for the subclass Lξ,μm,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 Lξ,μm,λ(δ,α,γ) satisfies the following conditions
(9)Re{Hξ,μm,λf(z)β(1+(1−β)(λ−ξ)μ+λ)mzβ}>γ|Hξ,μm,λf(z)β(1+(1−β)(λ−ξ)μ+λ)mzβ−1|+α.Where (μ>0,ξ,λ,γ≥0,m∈N0=N∪{0};β>0,z∈Ω) (see Ref. 6).
The above condition is equivalent to
(10)Hξ,μm,λf(z)β(1+(1−β)(λ−ξ)μ+λ)mzβ≺γ(z),where γ(z)=1+αz1−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
(11)|f́(z)(f(z)z)β−1|<1.If ξ=0 and m=λ=μ=1,L0,11,1(δ,α,γ)(f(z)β)
(12)f´(z)(f(z)z)β−1≺γ(z).γ(z)=1+z1−z,f´(z)≺1+z1−z.
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)={ξ1+ε2∫0z(h(v)−iε)v−(1+iξε1+ε2)g(v)ξ1+ε2dv}1+iεξ,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+c1z+c2z2+c3z3+…, z∈Ω belongs to the class A and μ∈C , then
|c2−μc12|≤2max{1,|2μ−1|}. The following functions can produce crisp results.8
h(z)=1+z21−z2andh(z)=1+z1−z,z∈Ω.
Let the function h(z) is analytic in Ω, with |h(z)|<1 and let
h(z)=c0+c1z+c2z2+c3z3+⋯.
|c0≤1|and|ck|≤1−|c0|2fork>0.
If h(z)=1+c1z+c2z2+c3z3+⋯,z∈Ω belongs to the class A then
|c2−γc12|≤{−4γ+2ifγ≤02if0≤γ≤14γ−2ifγ≥0.
For γ<0 or γ>1, equality holds if and only if h(z) equals 1+z1−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+z21−z2 or one of them is rotations. Inequality becomes equality when γ=0 if and only if h(z)=(1+λ2) 1+z1−z+(1−λ2) 1+z1−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 :
|c2−γc12|+γ|c1|2≤20<γ≤12,
|c2−γc12|+(1−γ)|c1|2≤212≤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.
Let (f(z))β∈Dβ which is given in Equation (6). If (f(z))β belongs to the class Lξ,μm,λ(δ,α,γ). Then
|a2(β)|≤α(1+δ)2α(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))m,|a3(β)|≤α(1+δ)2α(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m[1−(1−δα)A1+α(1+δ)A32A1]
|a4(β)|≤α(1+δ)2β(1+(λ−ξ)(−3−β)1+(λ−ξ)(1−β))m[1−(β−1)A3+2(1−δα)A22A2+3(β−1)(1−δα)A1A4+3α(β−1)(1+δ)A3A412A1A2−α(β−2)(1+δ)A3A412A1A2−312(1−2δα)]
A1=2β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))2m,A2=β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))m(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m,A3=(β−1)(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m,A4=α(1+δ)(1+(λ−ξ)(−3−β)1+(λ−ξ)(1−β))m.
If (f(z))β∈Lξ,μm,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)Hξ,μm,λf(z)β(1+(1−β)(λ−ξ)μ+λ)mzβ≺γ(w(z)),(15)γ(z)=1+αz1−δαz=1+α(1+δ)z+δα2(1+δ)z2+δ2α3(1+δ)z3+⋯.The definition of the function h(z) is:
h(z)=1+w(z)1−w(z)=1+c1h+c2h2+c3h3+⋯.
As may be seen from Equation (14), h∈A and
(16)w(z)=1−w(z)1+w(z)=12c1z+12(c2−12c12)z2+12(c3−c1c2+14c13)z3+⋯.Considering Equations (14), (15), and (16), we get at
(17)Hξ,μm,λf(z)β(1+(1−β)(λ−ξ)μ+λ)mzβ≺γ(w(z))=γ(z)(h(z)−1h(z)+1).=γ(12c1z+12(c2−12c12)z2+12(c3−c1c2+14c13)z3+⋯),=1+12α(1+δ)c1z+[12α(1+δ)(c2−12c12)+14δα2(1+δ)c12]z2++[12α(1+δ)((c3−c1c2+14c13)+12δα2(1+δ)(c2−12c12)c1+18δ2α2(1+δ)c12]…By use of the series expansion of
1+∑n=2∞(1+(λ−ξ)(β+n−1)1+((λ−ξ)(1−β))man(β)zn−1,
(18)Hξ,μm,λf(z)β(1+(1−β)(λ−ξ)μ+λ)mzβ=1+β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))ma2(β)z+[βa3(β)β(β−1)2!a22(β)](1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))mz2+[βa4(β)+β(β−1)a2(β)a3(β)+β(β−1)(β−2)3!a23(β)](1+(λ−ξ)(−3−β)1+(λ−ξ)(1−β))mz3+….Based on Equations (17) and (18), we compare the coefficients of z,z2 , and z3 in Equation (14) to obtain
(19)a2(β)=α(1+δ)c12α(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))m,(20)a3(β)≤α(1+δ)2α(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m[1−][c2−(1−δα)A1+α(1+δ)A32A1c12],(21)a4(β)=α(1+δ)2β(1+(λ−ξ)(−3−β)1+(λ−ξ)(1−β))m[c3−(β−1)A3+2(1−δα)A22A2c1c2+(3(β−1)(1−δα)A1A4+3α(β−1)(1+δ)A3A412A1A2−α(β−2)(1+δ)A3A412A1A2−312(1−2δα))c13].Given that h(z) has an analytic function and is confined to Ω ,
(22)|c0|≤1and|ck|≤1−|c0|2fork>0.By using Equation (22), we get
|a2(β)|≤α(1+δ)2α(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))m,|a3(β)|≤α(1+δ)2α(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m[1−(1−δα)A1+α(1+δ)A32A1]|a4(β)|≤α(1+δ)2β(1+(λ−ξ)(−3−β)1+(λ−ξ)(1−β))m[1−(β−1)A3+2(1−δα)A22A2+3(β−1)(1−δα)A1A4+3α(β−1)(1+δ)A3A412A1A2−α(β−2)(1+δ)A3A412A1A2−312(1−2δα)].
The proof is complete.
Using Theorem 1 and γ(z)=1+z1−z , we obtain the following.
Let (f(z))β∈Dβ which is given in Equation (6). If (f(z))β belonging to the class Lξ,μm,λ(1,1,γ). Then
|a2(β)|≤1α(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))m,
Based on Equations (19) and (20), we have
|a3(β)|≤1α(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m−A1α(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))mA3
|a4(β)|≤1β(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))m2A2+4(2β−1)(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))mA312A1A2]
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 Lξ,μ0,λ(1,1,γ).
Then
|a2(β)|≤1|a3(β)|≤1,
and
|a4(β)|≤1.
Let γ(z)=1+α(1+δ)z+δα2(1+δ)z2+…, where γ(z)∈D and γ´(z)>0. If (f(z))β which is given in Equation (6) belongs to the class Lξ,μm,λ(δ,α,γ) and φ∈ℕ, then
|a3(β)−φa22(β)|≤α(1+δ)β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))mmax{1,|(δα+δ)([2λ+β−1](1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m2β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))m)|}.a3(β)−φa22(β)=α(1+δ)c22β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m−[2βα(1+δ)(1−δα)(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))2m+(β−1)β2(1+δ)2(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m]8β2(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))2m(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))mc12−φβ2(1+δ)2c124α2(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))2m.
|a3(β)−φa22(β)|=α(1+δ)c22β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m[c2−vc12],
v=12[1−δβ+β(1+δ)((2φ+β−1)(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m2β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))2m)].
Using Lemma 2, we obtain
|a3(β)−φa22(β)|≤α(1+δ)β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))mmax{1,|δα+α(1+δ)([2λ+β−1](1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m2β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))m)|}.
The outcome for (f(z))β , which is provided as follows:
Hξ,μm,λf(z)β(1+(1−β)(λ−ξ)μ+λ)mzβ≺γ(z2),
Hξ,μm,λf(z)β(1+(1−β)(λ−ξ)μ+λ)mzβ≺γ(z).
The proof is complete.
The results of Lemma 4 are as follows:
Let γ(z)=1+α(1+δ)z+δα2(1+δ)z2+⋯,0≤δ≤1 and 0<β≤1. If
(f(z))β given by Equation (6) belongs to the class Lξ,μm,λ(δ,α,γ) , then τ∈ℂ
|a3(β)−φa22(β)|≤{α(1+δ)β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m[δα−α(1+δ)([2τ+β−1](1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m2β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))m)]ifτ≤τ1α(1+δ)β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m,ifτ1≤τ≤τ2α(1+δ)β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m[α(1+δ)([2τ+β−1](1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m2β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))m)−δα],ifτ≥τ2,
τ1=2α(δα−1)(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))2m−α(1+δ)(β−1)(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m2α(1+δ)(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m,τ2=2α(δα+1)(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))2m−α(1+δ)(β−1)(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m2α(1+δ)(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m.
Since (f(z))β∈Lξ,μm,λ(δ,α,γ) and γ(z) given by Equation (17), then a2(β) and a3(β) are given in Theorem 1. Furthermore,
|a3(β)−φa22(β)|=α(1+δ)2β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m[c2−tc12],
t=12[1−δ+α(1+δ)([2τ+β−1](1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m2β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))2m)].
Using Lemma 4, we can establish the inequality Equation (10) as follows.
|a3(β)−φa22(β)|≤α(1+δ)β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m[δα−α(1+δ)([2τ+β−1](1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m2β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))m)].
When 0≤t≤1, then τ1≤τ≤τ2 , and using Lemma 4, yields
|a3(β)−φa22(β)|≤α(1+δ)β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m.
Assuming t≥1, then τ2≤τ. Additionally, using Lemma 4, we get
|a3(β)−φa22(β)|≤α(1+δ)β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m[α(1+δ)([2τ+β−1](1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m2β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))m)−δα]
Using γ(z)=1+z1−z and Theorem 5, we arrive at the following conclusion:
Let γ(z)=1+2z+2z2+⋯.if(f(z))β which is given in Equation (6) belongs to the class Lξ,μm,λ(δ,α,γ), then for any real complex number σ
|a3(β)−φa22(β)|≤{2β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m[1−([2τ+β−1](1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m2β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))m)]ifτ≤τ12β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m,ifτ1≤τ≤τ22β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m[([2τ+β−1](1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m2β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))m)−1],ifτ≥τ2,
τ1=−(1−β)2(1+δ),τ2=4β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))2m−2(1−β)(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m4(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+2z+2z2+⋯.if(f(z))β given by Equation (6) belongs to the class
Lξ,μ0,λ(1,1,γ).
Then for any real complex number σ
|a2(β)−τa12(β)|≤{2−4τifτ≤τ1,2ifτ1≤τ≤4τ−2ifτ≥τ2,τ2,
The result of W. Ma. D. Minda [Ref. 8, Lemma 1] is obtained when τ1=0 and τ2=1 .
Let γ(z)=1+α(1+δ)z+δα2(1+δ)z2+⋯,0≤δ≤1 and 0<β≤1.If(f(z))β given by Equation (6) belongs to the class Lξ,μm,λ(δ,α,γ), and τ1≤τ3≤τ2. Let
τ3=12α(1+δ)(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m(2βδαβ(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))2m−α(1+δ)(1−β)(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m).
(23)≤β(1+δ)2β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m.(24)|a3(β)−τa22(β)|+2β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))2mα(1+δ)(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m×[1−δα+α(1+δ)([2τ+β−1](1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m2β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))m)]|a2(β)|2|a3(β)−τa22(β)|+2β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))2mα(1+δ)(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m×[1−δα+α(1+δ)([2τ+β−1](1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m2β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))m)]|a2(β)|2≤β(1+δ)2β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m|a3(β)−τa22(β)|+(τ−τ1)|a2(β)|2=(τ−12α(1+δ)(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m[2β(δα−1)(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))2m+α(1+δ)(β−1)(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m])×β2(1+δ)2(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))2m|c1|2+α(1+δ)2α(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m[c2−tc12]=α(1+δ)2α(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m[c2−tc12(β)+t|c1(β)|2].
It follows that using Lemma 4 yields
|a3(β)−τa22(β)|+(τ−τ1)|a2(β)|2≤α(1+δ)2α(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m,
This is the inequality of Equation (24). For
|a3(β)−τa22(β)|+(τ−τ1)|a2(β)|2=(2β(αδ+1)(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))2m2β(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))mβ(1+δ)(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m−τ)×α2(1+δ)24β(1+(λ−ξ)(−1−β)1+(λ−ξ)(1−β))2m|c1(β)|2+α(1+δ)2β(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m[c2−tc12]=α(1+δ)2β (1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m[|c2−tc12|+t|c1|2].
Using Lemma 4, the following results
|a3(β)−τa22(β)|+(τ−τ1)|a2(β)|2≤α(1+δ)2α(1+(λ−ξ)(−2−β)1+(λ−ξ)(1−β))m,
which is the inequality of Equation (24).
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.
This study did not involve human participants or animals, and therefore, ethical approval was not required.