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10.12688/f1000research.174492.1

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Study analytical function subordination properties by applying a novel linear operator

[version 1; peer review: 2 approved]

Majel

Maryam S.

Data Curation Formal Analysis Investigation Methodology Writing – Original Draft Preparation https://orcid.org/0009-0006-4216-7847 1 2 Hameed

Mustafa I.

Project Administration Supervision Writing – Original Draft Preparation Writing – Review & Editing https://orcid.org/0000-0003-0726-3180 a 1 2 1University of Anbar, Ramadi, Al Anbar Governorate, Iraq 2University of Anbar, Ramadi, Al Anbar Governorate, Iraq

a mustafa8095@uoanbar.edu.iq

No competing interests were disclosed.

31 12 2025

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23 12 2025

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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 study of theory for analytic univalent and multivalent functions is an old subject in mathematics, particularly in complex analysis, that has captivated a great deal of scholars owing to the sheer sophistication of its geometrical features as well as its many research possibilities. The study of univalent functions is one of many significant elements of complex analysis for both single and multiple variables. Investigators have become keen on the conventional investigation of this topic since at least 1907. Numerous scholars in the area of complex analysis have emerged since then, including Euler, Gauss, Riemann, Cauchy, and other people. Geometric function theory combines geometry and analysis.

Methods

This study employs the differential subordination technique to derive multiple characteristics from the new linear operator M σ , μ n , ς Υ ( s ) . The concept of the differential subordination subclass of analytical univalent functions is analyzed.

Results

In this section, We studied some results on differential subordination and superordination using a specific class of univalent functions stated on a specific space of univalent functions stated on the open unit disc. Using properties of the operator, we discovered a number of properties of superordinations and subordinations related to the idea of the Hadamard product. We investigated several aspects of superordinations and subordinations using a new operator M σ , μ n , ς Υ ( s ) .

Conclusions

A new operator M σ , μ n , ς Υ ( s ) : Λ ⟶ Λ has been established in this paper connected to the Dziok-Srivastava operator T σ n and the Hadamard product corresponding to the Komatu integral operator Ω μ ς . The difference operator M σ , μ n , ς ϒ ( s ) can have specific properties derived by applying the differential subordination technique. And the objective of this paper is to make use of the connection ( β 1 μ + 1 ) M σ , μ n + 1 , ς ϒ ( s ) = w ( M σ , μ n , ς ϒ ( s ) ) ′ + β 1 μ ( M σ , μ n , ς ϒ ( s ) ) .

Univalent Function Best Dominant Derivative Operator Differential Subordination Convex Function Hadamard Product.

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

1. Introduction

Gronwall’s Area Theorem, published in 1914, was a significant contribution to the theory of univalent functions. It is used for getting bounds on the coefficient values for meromorphic functions. Bieberbach resolved a similar issue regarding the class in 1916, as well as his known conjecture, which was mentioned in the exact same year but not applied until 1984, influenced the development of different methods in the geometric theory of complex variable functions. Estimates on the first two Taylor-Maclaurin coefficients are common in the study of bi-univalent functions since they are in the classes investigated by Gronwall and Bieberbach. Littlewood ¹ is the source of the fundamental findings on subordination that Rogosinski ² created and established. Subordination was recently utilized by Srivastava and Owa ³ to investigate the fascinating properties of the generalized hypergeometric function. A paper on differential subordinations, a generalization of differential disparities, was written by Miller and Mocanu. ⁴ The article is mainly concerned with the differential superordination of univalent functions in an open unit disk. To determine subordination properties, utilize the capabilities of the recently introduced operator M σ , μ n , ς ϒ ( s ) . Miller as well as Mocanu created the differential subordinations method in 1978, and the theory started to take shape in 1981. Refer to ^{5–
9} for further information. Suppose Λ indicates the type of functions of the given form ϒ ( s ) = s + ∑ η = 2 ∞ e η s η , e η ≥ 0 , (1) is analytic function in unit disk Δ ∗ = { s : s ∈ ℂ , | s | < 1 } . Allow ϕ in Λ be provided by ϕ ( s ) = s + ∑ η = 2 ∞ d η s η , d η ≥ 0 . (2)

The following is the Hadamard product of ϒ and ϕ : ( ϒ ∗ ϕ ) ( s ) = z + ∑ η = 2 ∞ e η d η s η , e η d η ≥ 0 . (3)

When ϒ and ϕ are analytic in Δ ∗ , the function ϒ is considered subservient to ϕ , expressed as ϒ ( s ) ≺ ϕ ( s ) ( s ∈ Δ ∗ ) .

The previously Schwarz function ω is present in Δ ∗ if and only if | w ( ω ) | < 1 as well as ω ( 0 ) = 0 , if and only if there exists the Schwarz function ω , in Δ ∗ , with ω ( 0 ) = 0 and | w ( ω ) | < 1 such that ϒ ( s ) = ϕ ( ω ( s ) ) , ( s ∈ Δ ∗ ) .

Additionally, we’re left with the following equivalency if ϕ is univalent in Δ ∗ by, ^{10–
13}: ϒ ≺ ϕ if and only if ϒ ( 0 ) = ϕ ( 0 ) and ϒ ( Δ ∗ ) ⊂ ϕ ( Δ ∗ ) .

2. Definitions and Lemmas

Definition 2.1.

¹⁴ The definition of the Komatu Integral operator Ω μ ς : Λ → Λ for ϒ ∈ Λ is Ω μ ς ( s ) = s + ∑ η = 2 ∞ ( μ + η − 1 μ ) − ς s n , ( μ > 0 ; ς ≥ 0 ) . (4)

Definition 2.2.

¹⁵ The definition of the Dziok-Srivastava operator Ω μ ς : Λ → Λ for ϒ ∈ Λ is T σ n ( β 1 , β 2 , . . , β n ; π 1 , π 2 . . , π σ ) ϒ ( s ) = s + ∑ η = 2 ∞ ( β 1 ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( μ + η − 1 μ ) e η s η , (5) where β n ∈ ℂ , n = 1 , 2 , 3 , … , n , π n ∈ ℂ ∖ { 0 , − 1 , − 2 , . . } , n = 1 , 2 , . . , σ .

Definition 2.3.

If a function ϒ ( s ) ∈ Λ and Hadamard products between operator T σ n as well as operator Ω μ ς given a new linear operator M σ , μ n , ς ϒ ( s ) : Λ → Λ , define M σ , μ n , ς ϒ ( s ) = T σ n ϒ ( s ) ∗ Ω μ ς ( s ) , ( s ∈ Δ ∗ ) , and M σ , μ n , ς ϒ ( s ) = s + ∑ η = 2 ∞ ( β 1 μ ς ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( 1 μ + η − 1 ) − ς + 1 e η s η . (6)

Form Eq. 6, we have ( β 1 μ + 1 ) M σ , μ n + 1 , ς ϒ ( s ) = w ( M σ , μ n , ς ϒ ( s ) ) ′ + β 1 μ ( M σ , μ n , ς ϒ ( s ) ) , ( s ∈ Δ ∗ ) (7)

It should be noted that M σ , μ n , ς ϒ has the following special cases. a)

The Komatu Integral operator Ω μ ς should be included if n = 1 , σ = 1 by. ¹⁴

Add the Dziok-Srivastava operator T σ n if ς = 0 ^{15,
16}

Salagean ¹⁷ examined the case if μ = 1 , ς = − n , n = 1 , σ = 1 .

It was examined by Salagean ¹⁷ and Flett ¹⁸ if μ = 1 , ς = n , n = 1 , σ = 1 .

Definition 2.4.

If σ ∈ ℕ , 0 ≤ π < 1 , μ > 0 ; n , ς ≥ 0 , then let R σ , μ n , ς ( π ) denote the class of a function ϒ ∈ Λ that satisfies the given conditions. R ( M σ , μ n , ς ϒ ( s ) ) ′ > π , ( s ∈ Δ ∗ ) . (8)

Lemma 2.1.

¹⁹ Assume that ϒ is in Λ , if R ( 1 + s ϒ ′ ′ ( s ) ϒ ′ ( s ) ) > − 1 2 , then 2 s ∫ 0 s ϒ ( τ ) dτ , ( s ∈ Δ ∗ and s ≠ 0 ) , is a convex functions.

Lemma 2.2.

²⁰ Let ϕ be a convex function in Δ ∗ such that

ψ ( s ) = ϕ ( s ) + ηβs ϕ ′ ( s ) , ( s ∈ Δ ∗ ) , in which β > 0 and η ∈ ℕ . If ϖ ( s ) = ϕ ( 0 ) + ϖ η s η + ϖ η + 1 s η + 1 + … , is analytic in Δ ∗ and ϖ ( s ) + βs ϖ ′ ( s ) ≺ ψ ( s ) , then ϖ ( s ) ≺ ϕ ( s ) .

Lemma 2.3.

¹⁹ Consider the following ψ ( 0 ) = e , ξ is analytic, univalent, and convex in Δ , γ ∈ ℂ ∖ { 0 } is a complex number that produces R { γ } ≥ 0 . Given ϖ ∈ Η [ e , η ] as well as ϖ ( s ) + s ϖ ′ ( s ) γ ≺ ψ ( s ) , then ϖ ( s ) ≺ ξ ( s ) ≺ ψ ( s ) , ( s ∈ Δ ∗ ) , where ξ ( s ) = γ η s γ η ∫ 0 s ψ ( τ ) τ γ η − 1 dτ , ( s ∈ Δ ∗ ) .

The optimal prevailing of the subordination is the convex function ξ .

3. Results and Discussion

In this study, we will derive multiple characteristics derived from the new linear operator M σ , μ n , ς ϒ ( s ) using differential subordination technique. Theorem 3.1.

Let ψ ( 0 ) = 1 , 0 ≤ π < 1 and ψ ( s ) = 1 + ( 2 π − 1 ) s 1 + s be convex in Δ ∗ . Suits the differential subordination ( M σ , μ n + 1 , ς ϒ ( s ) ) ′ ≺ ψ ( s ) for σ ∈ ℕ , μ > 0 ; n , ς ≥ 0 as well as ϒ ∈ A , then ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ξ ( s ) = ( 3 2 π − 1 ) + 3 / 2 ( β 1 μ + 1 ) ( 1 − π ) η s ( β 1 μ + 1 ) / η ζ ( ( β 1 μ + 1 ) / η ) , (9) where ζ ( x ) = ∫ 0 s τ x − 1 1 + τ dτ , ( s ∈ Δ ∗ ) .

Proof.

The result of differentiation Eq. (7) is ( β 1 μ + 1 ) ( M σ , μ n + 1 , ς ϒ ( s ) ) ′ = ( β 1 μ + 1 ) ( M σ , μ n , ς ϒ ( s ) ) ′ + s ( M σ , μ n , ς ϒ ( s ) ) ′ ′ ( M σ , μ n + 1 , ς ϒ ( s ) ) ′ = ( β 1 μ + 1 ) ( M σ , μ n , ς ϒ ( s ) ) ′ + s ( M σ , μ n , ς ϒ ( s ) ) ′ ′ ( β 1 μ + 1 ) ( M σ , μ n + 1 , ς ϒ ( s ) ) ′ = ( M σ , μ n , ς ϒ ( s ) ) ′ + s ( M σ , μ n , ς ϒ ( s ) ) ′ ′ ( β 1 μ + 1 ) . (10)

When Eq. (10) is used in Eq. (9), the subordination Eq. (9) is transformed into ( M σ , μ n , ς ϒ ( s ) ) ′ + s ( M σ , μ n , ς ϒ ( s ) ) ′ ′ ( β 1 μ + 1 ) ≺ ψ ( s ) = 1 + ( 3 2 π − 1 ) 1 + s . (11)

Let ϖ ( s ) = ( M σ , μ n , ς ϒ ( s ) ) ′ = ( s + ∑ η = 2 ∞ ( β 1 μ ς ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( 1 μ + η − 1 ) − ς + 1 e η s η ) ′ , ϖ ( s ) = 1 + ϖ 1 s + ϖ 2 s 2 + … , ϖ ∈ Η [ 1 , 1 ] , s ∈ Δ ∗ . (12)

Subordination is made possible in Eq. (11), through the use of Eq. (12). ϖ ( s ) + s ϖ ′ ( s ) ( β 1 μ + 1 ) ≺ ψ ( s ) = 1 + ( 3 / 2 π − 1 ) 1 + s .

Employing Lemma 2.3, has been ϖ ( s ) ≺ ξ ( s ) = ( β 1 μ + 1 ) η s ( β 1 μ + 1 ) ∕ η ∫ 0 s ψ ( τ ) τ ( ( β 1 μ + 1 ) / η ) − 1 dτ = ( β 1 μ + 1 ) η s ( β 1 μ + 1 ) ∕ η ∫ 0 s [ ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) 1 1 + τ ] τ ( ( β 1 μ + 1 ) / η ) − 1 1 + τ dτ = ( β 1 μ + 1 ) η s ( β 1 μ + 1 ) / η ∫ 0 s ( 3 / 2 π − 1 ) τ ( ( β 1 μ + 1 ) / η ) − 1 dτ + 3 / 2 ( 1 − π ) ( β 1 μ + 1 ) η s ( β 1 μ + 1 ) ∕ η ∫ 0 s τ ( ( β 1 μ + 1 ) / η ) − 1 1 + τ dτ = ( 3 / 2 π − 1 ) + 3 / 2 ( β 1 μ + 1 ) ( 1 − π ) η s ( β 1 μ + 1 ) ∕ η ζ ( ( β 1 μ + 1 ) ∕ η ) , then ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ξ ( s ) = ( 3 / 2 π − 1 ) + 3 / 2 ( β 1 μ + 1 ) ( 1 − π ) η s ( β 1 μ + 1 ) / η ζ ( ( β 1 μ + 1 ) / η ) .

Theorem 3.2.

Given that ξ is a convex function in Δ ∗ with ξ ( 0 ) = 1 and ψ ( s ) = ξ ( s ) + s ξ ′ ( s ) . If ϒ ∈ Λ , σ ∈ N , μ > 0 ; n , ς ≥ 0 , 0 ≤ π < 1 satisfies the subordination ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ψ ( s ) , then M σ , μ n , ς ϒ ( s ) s ≺ ξ ( s ) , ( s ∈ Δ ∗ ) . (13)

Proof.

Let ϖ ( s ) = M σ , μ n , ς ϒ ( s ) s ϖ ( s ) = s + ∑ η = 2 ∞ ( β 1 μ ς ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( 1 μ + η − 1 ) − ς + 1 e η s η s , ϖ ( s ) = 1 + ∑ η = 2 ∞ ( β 1 μ ς ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( 1 μ + η − 1 ) − ς + 1 e η s η − 1 ϖ ( s ) = 1 + ϖ η s η + ϖ η + 1 s η + 1 + … ( M σ , μ n , ς ϒ ( s ) ) ′ = ϖ ( s ) + s ϖ ′ ( s ) .

Consequently, with the help of the connection Eq. (13) grows ϖ ( s ) + s ϖ ′ ( s ) ≺ ψ ( s ) = ξ ( s ) + s ξ ′ ( s ) .

Utilizing Lemma 2.2, previously ϖ ( s ) ≺ ξ ( s ) .

Through the use of ϖ ( s ) = M σ , μ n , ς ϒ ( s ) s , we obtain M σ , μ n , ς ϒ ( s ) s ≺ ξ ( s ) .

Theorem 3.3.

If σ ∈ N , μ > 0 ; n , ς ≥ 0 as well as 0 ≤ π < 1 , then R σ , μ n + 1 , ς ( π ) ⊂ R σ , μ n , ς ( β ) such that ε = ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) ( β 1 μ + 1 ) η ζ ( ( β 1 μ + 1 ) / η ) , ζ = ∫ 0 s τ x − 1 1 + τ dτ . (14)

Proof.

Assume ϒ is in ϒ ∈ R σ , μ n + 1 , ς ( π ) . Next, based on Eq. (8), there is R ( M σ , μ n , ς ϒ ( s ) ) ′ > π this is equivalent to ( M σ , μ n + 1 , ς ϒ ( s ) ) ′ ≺ ψ ( s ) = 1 + ( 3 / 2 π − 1 ) s 1 + s .

Applying Theorem 3.1, we arrive at ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ξ ( s ) = ( 3 / 2 π − 1 ) + 3 / 2 ( β 1 μ + 1 ) ( 1 − π ) η s ( β 1 μ + 1 ) / η ζ ( ( β 1 μ + 1 ) / η ) .

Since ξ ( Δ ∗ ) is symmetric when compared to the real direction and ξ is convex, we are able to deduce that Re ( M σ , μ n , ς ϒ ( s ) ) ′ > Re ξ ( 1 ) = ε = ε ( π , β 1 , μ , η ) = ( 3 / 2 π − 1 ) + 3 / 2 ( β 1 μ + 1 ) ( 1 − π ) η ζ ( ( β 1 μ + 1 ) / η ) , where ξ ( 1 ) = ( β 1 μ + 1 ) η ∫ 0 1 [ 1 + ( 3 / 2 π − 1 ) τ 1 + τ ] τ ( ( β 1 μ + 1 ) / η ) − 1 dτ = ( β 1 μ + 1 ) η ∫ 0 1 [ ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) 1 1 + τ ] τ ( ( β 1 μ + 1 ) / η ) − 1 1 + τ dτ = ( β 1 μ + 1 ) η ∫ 0 1 ( 3 / 2 π − 1 ) τ ( ( β 1 μ + 1 ) / η ) − 1 dτ + 3 / 2 ( 1 − π ) ( β 1 μ + 1 ) η ∫ 0 1 τ ( ( β 1 μ + 1 ) / η ) − 1 1 + τ dτ = ( 3 / 2 π − 1 ) + 3 / 2 ( β 1 μ + 1 ) ( 1 − π ) η ζ ( ( β 1 μ + 1 ) / η )

It leads us to the conclusion that R σ , μ n + 1 , ς ( π ) ⊂ R σ , μ n , ς ( β ) . The evidence is finished.

Theorem 3.4.

Let ψ be a convex function in Δ ∗ , ψ ( 0 ) = 1 , 0 ≤ π < 1 , and ψ ( s ) = 1 + ( 3 / 2 π − 1 ) s 1 + s . If σ ∈ ℕ , μ > 0 ; n , ς ≥ 0 , ϒ ∈ Λ satisfies the subordination ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ψ ( s ) , then M σ , μ n , ς ϒ ( s ) s ≺ ξ ( s ) = ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) ln ( 1 + s ) s , ( s ∈ Δ ∗ ) .

The function ξ is convex as well as the most prevailing.

Proof.

Suppose ϖ ( s ) = M σ , μ n , ς ϒ ( s ) s = s + ∑ η = 2 ∞ ( β 1 μ ς ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( 1 μ + η − 1 ) − ς + 1 e η s η s , = ϖ ( s ) = 1 + ϖ η s η + ϖ η + 1 s η + 1 + … . (15)

By Eq. (15) in relation to ϖ , we get ( M σ , μ n , ς ϒ ( s ) ) ′ = ϖ ( s ) + s ϖ ′ ( s ) , ( s ∈ Δ ∗ ) . (16)

By applying Eq. (16), differential subordination ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ψ ( s ) , is obtained ϖ ( s ) + s ϖ ′ ( s ) ≺ ψ ( s ) = 1 + ( 3 / 2 π − 1 ) s 1 + s , ( s ∈ Δ ∗ ) .

Utilizing Lemma 2.3, has been ϖ ( s ) ≺ ξ ( s ) = 1 s ∫ 0 s ψ ( τ ) dτ = 1 s ∫ 0 s ( 1 + ( 3 / 2 π − 1 ) τ 1 + τ ) dτ = 1 s ∫ 0 s [ ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) 1 1 + τ ] dτ = 1 s ∫ 0 s ( 3 / 2 π − 1 ) dτ + 3 / 2 ( 1 − π ) s ∫ 0 s 1 1 + τ dτ = ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) ln ( 1 + s ) s .

With the help of Eq. (15), we had M σ , μ n , ς ϒ ( s ) s ≺ ξ ( s ) = ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) ln ( 1 + s ) s .

Theorem 3.4, has been fully proved.

Theorem 3.5.

If ψ is convex function in Δ ∗ and ψ ( 0 ) = 1 , 0 ≤ π < 1 , ψ ( s ) = 1 + ( 3 / 2 π − 1 ) s 1 + s , ϒ ∈ Λ , σ ∈ ℕ , κ > 0 ; n , ς ≥ 0 satisfies the subordination 1 − M σ , μ n , ς ϒ ( s ) ( M σ , μ n , ς ϒ ( s ) ) ′ ′ [ ( M σ , μ n , ς ϒ ( s ) ) ′ ] 2 ≺ ψ ( s ) , (17) then M σ , μ n , ς ϒ ( s ) s ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ξ ( s ) = ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) ln ( 1 + s ) s .

Proof.

Suppose ϖ ( s ) = M σ , μ n , ς ϒ ( s ) s ( M σ , μ n , ς ϒ ( s ) ) ′ , ϖ ( s ) = s + ∑ η = 2 ∞ ( β 1 μ ς ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( 1 μ + η − 1 ) − ς + 1 e η s η s ( 1 + ∑ η = 2 ∞ ( β 1 μ ς ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( 1 μ + η − 1 ) − ς + 1 e η s η − 1 ) , so that with the help of Eq. (17) becomes ϖ ( s ) + s ϖ ′ ( s ) ≺ ψ ( s ) = 1 + ( 3 / 2 π − 1 ) s 1 + s , ( s ∈ Δ ∗ ) .

By Lemma 2.3 allows us to have ϖ ( s ) ≺ ξ ( s ) = 1 s ∫ 0 s ψ ( τ ) dτ = 1 s ∫ 0 s ( 1 + ( 3 / 2 π − 1 ) τ 1 + τ ) dτ = 1 s ∫ 0 s [ ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) 1 1 + τ ] dτ = ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) ln ( 1 + s ) s , then M σ , μ n , ς ϒ ( s ) s ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ξ ( s ) = ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) ln ( 1 + s ) s .

Theorem 3.6.

If ϒ ∈ Λ and suppose that ξ is an analytic function that fulfills the next inequality R { 1 + s ψ ′ ′ ( s ) ψ ′ ( s ) } > − 1 2 , and ψ ( 0 ) = 1 , ψ ′ ( 0 ) ≠ 0 , σ ∈ N , μ > 0 ; n , ς ≥ 0 satisfies the subordination ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ψ ( s ) , ( s ∈ Δ ∗ ) , (18) then M σ , μ n , ς ϒ ( s ) s ≺ ξ ( s ) = 1 s ∫ 0 s ψ ( τ ) dτ .

Proof.

Suppose ϖ ( s ) = M σ , μ n , ς ϒ ( s ) s = s + ∑ η = 2 ∞ ( β 1 μ ς ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( 1 μ + η − 1 ) − ς + 1 e η s η s , ϖ ( s ) = 1 + ϖ η s η + ϖ η + 1 s η + 1 + … . (19)

By Eq. (19) in relation to w, we get ( M σ , μ n , ς ϒ ( s ) ) ′ = ϖ ( s ) + s ϖ ′ ( s ) , ( s ∈ Δ ∗ ) . (20)

Subordination Eq. (18), when applied to Eq. (20), grows ϖ ( s ) + s ϖ ′ ( s ) ≺ ψ ( s ) .

By Lemma 2.3 allows us to have ϖ ( s ) ≺ ξ ( s ) , in other words, M σ , μ n , δ ϒ ( s ) s ≺ ξ ( s ) = 1 s ∫ 0 s ψ ( τ ) dτ .

Theorem 3.7.

If ϒ ∈ Λ and suppose ξ be convex function in Δ ∗ , ξ ( 0 ) = 1 and ψ ( s ) = ξ ( s ) + s ξ ′ ( s ) , σ ∈ N , μ > 0 ; n , ς ≥ 0 , 0 ≤ π < 1 satisfies subordination 1 − M σ , μ n , ς ϒ ( s ) ( M σ , μ n , ς ϒ ( s ) ) ′ ′ [ ( M σ , μ n , ς ϒ ( s ) ) ′ ] 2 ≺ ψ ( s ) , ( s ∈ Δ ∗ ) , (21) then M σ , μ n , ς ϒ ( s ) s ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ξ ( s ) . (22)

Proof.

Suppose ϖ ( s ) = M σ , μ n , ς ϒ ( s ) s ( M σ , μ n , ς ϒ ( s ) ) ′ , p ∈ H [ 1 , 1 ] .

Taking the product of both perspectives gives us ϖ ( s ) + s ϖ ′ ( s ) = 1 − M σ , μ n , ς ϒ ( s ) ( M σ , μ n , ς ϒ ( s ) ) ′ ′ [ ( M σ , μ n , ς ϒ ( s ) ) ′ ] 2 , so that with the help of Eq. (21) becomes ϖ ( s ) + s ϖ ′ ( s ) ≺ ψ ( s ) = ξ ( s ) + s ξ ′ ( s ) , ( s ∈ Δ ∗ ) .

By Lemma 2.2, we obtain ϖ ( s ) ≺ ξ ( s ) , then M σ , μ n , ς ϒ ( s ) s ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ξ ( s ) .

4. Conclusions

There are many fascinating discoveries regarding harmonic multivalent functions derived from differential operators. The study concentrated on a subclass of analytical univalent functions related to the concept of differential subordination. Everyone looked at a few differential subordination and superordination outcomes, such as a class determined by a dimension for univalent meromorphic functions inside the open unit disc. Learn about geometrical characteristics such as coefficient border, coefficient disparities, distortion theorem, closing theorem, severe points, starlikeness radii, convexity, near-perfect convexity, and combining principles. An analysis is conducted on the concept of the differential subordination subclass of analytical univalent functions. Utilizing a particular class of univalent functions stated on a particular space of univalent functions stated on the open unit disc, we examined certain findings on differential subordination as well as superordination. We found several properties of subordinations as well as superordinations connected to the notion of Hadamard product by using characteristics of the operator. We examined various facets of subordinations as well as superordinations through a novel operator, M σ , μ n , ς ϒ ( s ) .

Data availability

Data sharing is not applicable to this article, as no data were created or analyzed in this study.

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10.5256/f1000research.192398.r447074

Reviewer response for version 1

A. Jassim

Kassim

1 Referee 1University of Baghdad, Baghdad, Baghdad Governorate, Iraq

Competing interests: No competing interests were disclosed.

16 1 2026

2026

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

recommendation

approve

The paper is studied some results on differential subordination and superordination using a class of univalent functions defined on the open unit disc. Many properties of the of subordinations and superordinations are investigated , using new operator and it's identity .Also , cited references is really used in paper and many of these are modern which make paper more specified and important.

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

Yes

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

Yes

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

Yes

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

Yes

Are the conclusions drawn adequately supported by the results?

Yes

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

Yes

Reviewer Expertise:

Complex analysis

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

10.5256/f1000research.192398.r447076

Reviewer response for version 1

Lupas

Alina Alb

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

Competing interests: No competing interests were disclosed.

7 1 2026

2026

recommendation

approve

REVIEWER’s REPORT

On the paper

Study analytical function subordination properties by applying a novel linear operator

by Maryam S. Majel and Mustafa I. Hameed

In this paper the authors define and study a new operator by using the Dziok-Srivastava operator and the Hadamard product with the Komatu integral operator. They establish several properties of subordinations and superordinations regarding the new defined operator.

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

The paper doesn’t require a revision.

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

"No competing interests were disclosed"

4.01.2026

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

Yes

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

Not applicable

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

Yes

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

Yes

Are the conclusions drawn adequately supported by the results?

Yes

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

Yes

Reviewer Expertise:

Complex analysis

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