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Article

# Properties of Differential Subordination and Superordination for Multivalent Functions Associated with the Convolution Operators

by
Luminiţa-Ioana Cotîrlă
1,* and
Abdul Rahman S. Juma
2
1
Department of Mathematics, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
2
Department of Mathematics, College of Education for Pure Sciences, University of Anbar, Ramadi 31001, Iraq
*
Author to whom correspondence should be addressed.
Axioms 2023, 12(2), 169; https://doi.org/10.3390/axioms12020169
Submission received: 8 December 2022 / Revised: 1 February 2023 / Accepted: 3 February 2023 / Published: 7 February 2023
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

## Abstract

:
Using convolution (or Hadamard product), we define the El-Ashwah and Drbuk linear operator, which is a multivalent function in the unit disk , and satisfy its specific relationship to derive the subordination, superordination, and sandwich results for this operator by using properties of subordination and superordination concepts.
MSC:
30C45

## 1. Introduction and Definitions

The set $Ω U$ denotes the class of all analytic functions in the open unit disk and as the subclass of $Ω U$, which consists of the form functions
With $A p$ as the class of all multivalent functions in open unit disk $U$ of the form
Additionally, we use $A = A 1$ to denote the class of analytic functions in the open unit disk $U$ and normalize them with $f$(0) = 0, $f ′$(0) = 1.
Additionally, consider $S$ as the class of the univalent function in $U$,
Let and $K$ be the subclasses of $A$ such that:
then $f$ is a starlike function;
, then $f$ is a convex function;
$f ∈ K : R e f 1 ′ w g ′ w > 0 : g ∈ C , w ∈ U$, then $f$ is a close-to-convex function.
If the functions $f$ and $g$ are analytic in $U$, then we say $f$ is subordinate to $g$ or f is said to be superordinate to f in $U$, written as $f ≺ g$ or $f w ≺ g w$ if there is a Schwarz function $υ w$ analytic in $U$, with $υ w < 1$, so that $f w = g υ w$ and $w ∈ U$. In particular, if the function g is univalent in $U$, then the subordination $f ≺ g$ is equivalent to $f 0 = g 0$ and $f U ⊂ g U$, (see [1,2,3,4,5,6,7,8]).
If $f , g ∈ A p$, where $f w$ is provided by (1) and $g w$ is defined by
$g w = w p + ∑ k = 1 + p ∞ a k w k , w ∈ U ,$
the Hadamard product (or convolution) of the function is defined by
$f w × g w = w p + ∑ k = 1 + p ∞ a k b k w k , w ∈ U = f × g w .$
Let such that $R e c − a ≥ 0$ and
El-Ashwah and Drbuk [5] introduced the linear operator defined by
It is readily verified from (4) that
Putting $a = c$ in (4), we obtain the Prajapat operator $J p n θ , λ$, see [9].
Additionally, when $n = 0$, we obtain the Erdelyi-Kober integral operator $I p , δ a , c$, see [10].
Definition 1.
Let$Υ$:$₵ 3 × U → ₵$and$h$ ($w )$ be univalent in $U$. If $p$ ($w$) is analytic in $U$, that fulfils the second-order differential subordination [11]:
then $p w$ is the differential subordination solution of (6).
Definition 2.
Let$Υ 1$:$₵ 3 × U → ₵$and$h$($w )$be univalent in$U$. If$p$($w$) andare univalent in$U$and$p w$fulfill the second-order differential superordination [11]:
then$p w$is the differential superordination solution of (7).
Definition 3.
Let$Q$be the collections of functions$f$that are analytic and injective on$U ¯ ∖ E f$, when and$f ′ w ≠ 0$for[11].
Lemma 1.
Let$p 1 w$be the univalent function in$U$and let$Σ$and$ϑ$be holomorphic in a domain$p 1$($U ) ⊂ D ,$with$ϑ ω ≠ 0$, when$ω ∈ p 1$ ($U$). Set and $ℏ w = Σ ( p 1 w + O w$. Suppose that [12]
$i$$O$is starlike in$U$.
,.
If$p 2 w$is holomorphic in$U$with$p 2 0 = p 1 0$,$p 2$($U ) ⊂ D$, and$Σ p 2 w + w p 2 ′ w ϑ p 2 w ≺ Σ p 1 w + w p 1 ′ w ϑ p 1 w$, then.
Lemma 2.
Let$p 1 w$be convex in$U$and$β 1 ∈ ₵ , β 2 ∈ ₵ *$with$R e 1 + p 1 ″ w p 1 ′ w > max 0 , − R e β 1 β 2$. If$p 2 w$is holomorphic in$U$and$β 1 p 2 w + β 2 w p 2 ′ w ≺ β 1 p 1 w + β 2 w p 1 ′ w ,$then[11].
Lemma 3.
Let$p 1 w$be convex univalent in$U$and let$Σ$and$ϑ$be holomorphic in a domain$D$,$p 1$ ($U ) ⊂ D$. Suppose that [12]
$i$$w p 1 ′ w ϑ ( p 1 w$is starlike univalent in$U$.
,$w ∈ U$.
If, with$p 2$ ($U ) ⊂ D$, $Σ p 2 w + w p 2 ′ w ϑ ( p 2 w$ is univalent in $U$ and $Σ p 1 w + w p 1 ′ w ϑ p 1 w ≺ Σ p 2 w + w p 2 ′ w ϑ p 2 w$, then .
Lemma 4.
Let$p 1 w$be convex in$U$and$R e β > 0$.
If,$p 2 w + β w p 2 ′ w$is univalent in$U$and$p 1 w + β w p 1 ′ w ≺ p 2 w + β w p 2 ′ w$, then$p 1 w ≺ p 2 w$[12].

## 2. Subordination Results

Theorem 1.
Let$b w$be convex univalent in$U$, with$b 0 = 1$, $a 1 > 0 , 0 ≠ a 2 ∈ ₵$and suppose
$R e 1 + b ″ w b ′ w > max 0 , − R e a 1 a 2 .$
If$f ∈ A$, it satisfies the subordination:
then
Proof.
Consider
Then
We have
By using (5) we obtain
and
By using the hypothesis, we obtain $q w + a 2 a 1 w q ′ w ≺ b w + a 2 a 1 w b ′ w$.
Additionally, apply Lemma 2, when $β 1 = 1$ and $β 2 = a 2 a 1$, then
Corollary 1.
Let$b w$be convex univalent in$U$, with$b 0 = 1$, $a 1 > 0 , 0 ≠ a 2 ∈ ₵$and suppose
$R e 1 + b ″ w b ′ w > max 0 , − R e a 1 a 2 .$
If$f ∈ A$, it satisfies the subordination:
$1 − a 2 p + λ θ J p n θ , λ f w w p a 1 + a 2 p + λ θ J p n + 1 θ , λ f w J p n θ , λ f w a 1 J p n θ , λ f w w p a 1 ≺ b w + a 2 a 1 w b ′ w ,$
then
$J p n θ , λ f w w p a 1 ≺ b w .$
Theorem 2.
Let$b$be convex univalent in,$b 0 = 1$, and$b w ≠ 0$for all$w ∈ U$, and suppose that$b$satisfies:
$R e p + w t σ w p a 2 + w ε σ + 1 w p a 2 w + σ − 1 w b ′ w b w + w b ″ w b ′ w > 0 ,$
where$σ$, $ε , t ∈ ₵ , a 1 > 0 , 0 ≠ a 2 ∈ ₵$and$w ∈ U$. Suppose that$w p b w σ − 1 b ′ w$is a starlike univalent in$U$.
If$f ∈ A$satisfies the subordination:
where
then
Proof.
Let $H β = t + ε β β σ$ and $L β = a 2 β σ − 1$, $0 ≠ β ∈ ₵$, when $H β$ and $L β$ are analytic in $₵$. □
Then, we obtain $G w = w b ′ w L b w = a 2 w p b w σ − 1 b ′ w$ and $y w = H b w + G w = t + ε b w b w σ + a 2 w p b w σ − 1 b ′ w$.
Since $w p b w σ − 1 b ′ w$ is starlike, then $G w$ is starlike in $U ,$ and
$R e y ′ w G w = R e p + w t σ w p a 2 + w ε σ + 1 w p a 2 w + σ − 1 w b ′ w b w + w b ″ w b ′ w > 0$
.
Then,
We obtain
Since
That
From (8) we obtain and using Lemma 1 we obtain $q w ≺ b w .$
Corollary 2.
Let$b$be convex univalent in,$b 0 = 1$, and$b w ≠ 0$for all$w ∈ U$, and suppose that$b$satisfies:
$R e p + w t σ w p a 2 + w ε σ + 1 w p a 2 w + σ − 1 w b ′ w b w + w b ″ w b ′ w > 0 ,$
where$σ ,$$ε , t ∈ ₵ , a 1 > 0 , 0 ≠ a 2 ∈ ₵$and$w ∈ U$.
Suppose that$w p b w σ − 1 b ′ w$is a starlike univalent in$U$.
If$f ∈ A$, it satisfies the subordination:
then
$p + λ θ J p n + 1 θ , λ f w + 1 − p + λ θ J p n θ , λ f w w p a 1 ≺ b w .$

## 3. Superordination Results

Theorem 3.
Let$b w$be convex in$U$, with$b 0 = 1$, $a 1 > 0$, $R e a 2 > 0$, if$f ∈ A$,
and
is univalent in$U$and satisfies the superordination.
then
Proof.
Consider
then
We have
with the same steps of Theorem 1 and using the hypothesis, we obtain
Apply Lemma 4 we obtain
Corollary 3.
Let$b w$be convex in$U$, with$b 0 = 1$, $a 1 > 0$, $R e a 2 > 0$, if$f ∈ A$,
$J p n θ , λ f w w p a 1 ∈ Ω q 0 , 1 ∩ Q$
and
$1 − a 2 p + λ θ J p n θ , λ f w w p a 1 + a 2 p + λ θ J p n + 1 θ , λ f w J p n θ , λ f w a 1 J p n θ , λ f w w p a 1$
is univalent in$U$and satisfies the superordination
$b w + a 2 a 1 w b ′ w ≺ 1 − a 2 p + λ θ J p n θ , λ f w w p a 1 + a 2 p + λ θ J p n + 1 θ , λ f w J p n θ , λ f w a 1 J p n θ , λ f w w p a 1 ,$
then
$b w ≺ J p n θ , λ f w w p a 1 .$
Theorem 4.
Let$b$be convex univalent in$b 0 = 1$and$b w ≠ 0$for all$w ∈ U$and suppose that$b$satisfies:
where,$ε , t ∈ ₵ , 0 ≠ a 2 ∈ ₵ *$, $w ∈ U$, and$w b w σ − 1 b ′ w$are all starlike univalent in$U$.
If$f ∈ A$, satisfies the condition:
and$M p , n , λ , θ , ε , a 1 , a 2 ; w$is univalent in$U$.
If, then
Proof.
Let $H β = t + ε β β σ$ and $L β = a 2 β σ − 1$, $0 ≠ β ∈ ₵$, when $H β$ is analytic in $₵$ and $L β ≠ 0$ is analytic in $₵ ∕ 0$. Then, we obtain $G w = w p b ′ w L b w = a 2 w p b w σ − 1 b ′ w$. □
Since $w p b w σ − 1 b ′ w$ is starlike, then $G w$ is starlike in $U ,$ and
$R e H ′ b w L b w = R e t + ε b w b w σ ′ a 2 b w σ − 1 = R e t σ a 2 b ′ w + ε σ + 1 a 2 b w b ′ w > 0 ;$
Now, let
From (8) we obtain
Using Lemma 3 we obtain $b w ≺ q w .$
Corollary 4.
Let$b$be convex univalent in U,$b 0 = 1$, and$b w ≠ 0$for all$w ∈ U ,$and suppose that$b$satisfies:
$R e t σ a 2 b ′ w + ε σ + 1 a 2 b w b ′ w , > 0 ,$
where,$ε , t ∈ ₵ , 0 ≠ a 2 ∈ ₵ *$, $w ∈ U$, and$w b w σ − 1 b ′ w$are starlike univalent in$U$.
Let$f ∈ A$, satisfies the condition:
and$M p , n , λ , θ , ε , a 1 , a 2 ; w$is univalent in$U$.
If, then
$b w ≺ p + λ θ J p n + 1 θ , λ f w + 1 − p + λ θ J p n θ , λ f w w p a 1 .$

## 4. Sandwich Results

By combining the above theories, we obtain the following two sandwich theories.
Theorem 5.
Let$b 1 , b 2$be convex univalent in$U$, with$b 1 0 = b 2 0 = 1$$R e a 2 > 0$and
$R e 1 + q ″ w q ′ w > max 0 , − R e a 1 a 2 ,$
where$a 1 > 0 , 0 ≠ a 2 ∈ ₵$.
If$f ∈ A$and
and
is univalent in$U$, it satisfies:
then.
Theorem 6.
Let$b 1 , b 2$be convex univalent in$U$, with, and let$f ∈ A$satisfy the condition:
and$M p , n , λ , θ , ε , a 1 , a 2 ; w$is univalent in$U$.
If
then

## 5. Conclusions

In this paper, using the convolution (or Hadamard product) we defined the El-Ashwah and Drbuk linear operator, which is a multivalent function in the unit disk U and satisfied its specific relationship to derive the subordination, superordination, and some sandwich results for this operator using the properties of subordination and superordination concepts. The interesting results can be obtained for other operators using the same techniques of subordinations and superordinations.

## Author Contributions

Conceptualization, L.-I.C. and A.R.S.J. methodology, L.-I.C. and A.R.S.J. software, L.-I.C. and A.R.S.J.; validation, L.-I.C. and A.R.S.J.; formal analysis, L.-I.C. and A.R.S.J.; investigation, L.-I.C. and A.R.S.J.; resources, L.-I.C. and A.R.S.J.; data curation, L.-I.C. and A.R.S.J.; writing—original draft preparation, L.-I.C. and A.R.S.J.; writing—review and editing, L.-I.C. and A.R.S.J.; visualization, L.-I.C. and A.R.S.J.; supervision, L.-I.C. and A.R.S.J.; project administration, L.-I.C. and A.R.S.J.; funding acquisition, L.-I.C. All authors have read and agreed to the published version of the manuscript.

## Funding

This research received no external funding.

Not applicable.

## Conflicts of Interest

The authors declare no conflict of interest.

## References

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MDPI and ACS Style

Cotîrlă, L.-I.; Juma, A.R.S. Properties of Differential Subordination and Superordination for Multivalent Functions Associated with the Convolution Operators. Axioms 2023, 12, 169. https://doi.org/10.3390/axioms12020169

AMA Style

Cotîrlă L-I, Juma ARS. Properties of Differential Subordination and Superordination for Multivalent Functions Associated with the Convolution Operators. Axioms. 2023; 12(2):169. https://doi.org/10.3390/axioms12020169

Chicago/Turabian Style

Cotîrlă, Luminiţa-Ioana, and Abdul Rahman S. Juma. 2023. "Properties of Differential Subordination and Superordination for Multivalent Functions Associated with the Convolution Operators" Axioms 12, no. 2: 169. https://doi.org/10.3390/axioms12020169

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