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Article

# Subclasses of p-Valent Functions Associated with Linear q-Differential Borel Operator

by
1,*,†,
Emilia-Rodica Borşa
1,† and
Sheza M. El-Deeb
2,3,†
1
Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania
2
Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt
3
Department of Mathematics, College of Science and Arts, Al-Badaya, Qassim University, Buraidah 52571, Saudi Arabia
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2023, 11(7), 1742; https://doi.org/10.3390/math11071742
Submission received: 15 February 2023 / Revised: 22 March 2023 / Accepted: 31 March 2023 / Published: 5 April 2023
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications)

## Abstract

:
The aim of the present paper is to introduce and study some new subclasses of p-valent functions by making use of a linear q-differential Borel operator.We also deduce some properties, such as inclusion relationships of the newly introduced classes and the integral operator $J μ , p$.
MSC:
05A30; 30C45; 11B65; 47B38

## 1. Introduction

Let $A p$ denote the class of functions of the form:
$F ( ς ) = ς p + ∑ j = p + 1 ∞ a j ς j ( p ∈ N = { 1 , 2 , . . . } ) ,$
which are analytic in the open unit disc $Δ = ς ∈ C : ς < 1 .$
Let $P p , k α$ be the class of functions $h ( ς )$ analytic in $Δ$ satisfying the properties $h ( 0 ) = p$ and
$∫ 0 2 π ℜ h ( ς ) − α p − α d θ ≤ k π ,$
where $ς = r e i θ , k ≥ 2$ and $0 ≤ α < p .$ This class was introduced by (Aouf [1] with $λ = 0$).
We note that
(i)
$P 1 , k α = P k α$$k ≥ 2 , 0 ≤ α < 1$ (see Padmanabhan and Parvatham [2]);
(ii)
$P 1 , k 0 = P k k ≥ 2$(see Pinchuk [3] and Robertson [4]);
(iii)
$P p , 2 α = P p , α 0 ≤ α < p , p ∈ N ,$ where $P p , α$ is the class of functions with a positive real part greater than $α$(see [1]);
(iv)
$P p , 2 0 = P p p ∈ N ,$ where $P p$ is the class of functions with a positive real part (see [1]).
From (2), we have $h ( ς ) ∈ P p , k α$ if and only if there exists $h 1 , h 2 ∈ P p α$ such that
$G ( ς ) = k 4 + 1 2 h 1 ( ς ) − k 4 − 1 2 h 2 ( ς ) ς ∈ Δ .$
For two functions $F ( ς )$ given by (1) and $H ( ς )$ given by
$H ( ς ) = ς p + ∑ j = p + 1 ∞ b j ς j$
the Hadamard product (or convolution) is defined by
$( F ∗ H ) ( ς ) = ς p + ∑ j = p + 1 ∞ a j b j ς j = ( H ∗ F ) ( ς ) .$
Define here a Borel distribution with parameter $λ$, which is a discrete random variable denoted by $χ$. This variable takes the values $1 , 2 , 3 , . . .$ with the probabilities $e − λ 1 ! , 2 λ e − 2 λ 2 ! , 9 λ 2 e − 3 λ 3 ! , . . .$, respectively.
Wanas and Khuttar [5] recently introduced the Borel distribution (BD) whose probability mass function is (see [6,7])
$P ( χ = ρ ) = ρ λ ρ − 1 e − λ ρ ρ ! , ρ = 1 , 2 , 3 , . . .$
Wanas and Khuttar studied a series $M ( λ ; ς )$ whose coefficients are probabilities of the Borel distribution (BD)
$M p ( λ ; ς ) = ς p + ∑ j = p + 1 ∞ λ j − p j − p − 1 e − λ j − p j − p ! ς j , 0 < λ ≤ 1 , = ς p + ∑ j = p + 1 ∞ ϕ j , p ( λ ) ς k , 0 < λ ≤ 1 ,$
where
$ϕ j , p ( λ ) = λ j − p j − p − 1 e − λ j − p j − p !$
We propose a linear operator $D ( p , λ ; ς ) F : A p → A p$ as follows
$D ( p , λ ; ς ) F ( ς ) = M p ( λ ; ς ) ∗ F ( ς ) = ς p + ∑ j = p + 1 ∞ λ j − p j − p − 1 e − λ j − p j − p ! a j ς j , 0 < λ ≤ 1 .$
In a recent paper, Srivastava [8] studied various types of operators regarding q-calculus. We recall further some important definitions and notations. The q-shifted factorial is defined for $λ , q ∈ C$ and $n ∈ N 0 = N ∪ { 0 }$ as follows
By using the q-gamma function $Γ q ( ς ) ,$ we get
$q μ ; q j = 1 − q j Γ q μ + j Γ q μ , j ∈ N 0 ,$
where (see [9])
$Γ q ( ς ) = 1 − q 1 − ς q ; q ∞ q ς ; q ∞ , q < 1 .$
Furthermore, we note that
$μ ; q ∞ = ∏ j = 0 ∞ 1 − μ q j , q < 1 ,$
and, the q-gamma function $Γ q ( ς )$ is known
$Γ q ( ς + 1 ) = j q Γ q ( ς ) ,$
where $j q$ denotes the basic q-number defined as follows
$[ j ] q : = 1 − q j 1 − q , j ∈ C , 1 + ∑ i = 1 j − 1 q i , j ∈ N .$
Using the definition from (7), we have the next two products:
(i)
For a non negative integer j, the q-shifted factorial is defined by
$[ j ] q ! : = 1 , if j = 0 , ∏ n = 1 j [ n ] q , if j ∈ N .$
(ii)
For a positive number r, the q-generalized Pochhammer symbol is given by
$r q , j : = 1 , if j = 0 , ∏ n = r r + j − 1 [ n ] q , if j ∈ N .$
In terms of the classical (Euler’s) gamma function $Γ ς$, we have
$Γ q ς → Γ ς as q → 1 − .$
Furthermore, we notice that
$lim q → 1 − q μ ; q j 1 − q j = μ j .$

## 2. Preliminaries

In order to establish our new results, we have to recall the construct of a q-derivative operator. Considering $0 < q < 1$, the q-derivative operator [10] (see also other specific and generalized results [11,12,13,14,15]) for $D ( p , λ ; ς ) F$ is defined by
$D q D ( p , λ ; ς ) F ( ς ) : = D ( p , λ ; ς ) F ( ς ) − D ( p , λ ; ς ) F ( q ς ) ς ( 1 − q ) = p q ς p − 1 + ∑ j = p + 1 ∞ [ j ] q λ j − p j − p − 1 e − λ j − p j − p ! a j ς j − 1 ,$
where $[ j ] q$ is defined in (7)
For $α > − 1$ and $0 < q < 1$, we obtain the linear operator $D p , λ μ , q F : A p → A p$ by
$D p , λ μ , q F ( ς ) ∗ N p , μ + 1 q ( ς ) = ς p q D q D ( p , λ ; ς ) F ( ς ) , ς ∈ Δ ,$
where the function $N p , α + 1 q$ is given by
$N p , μ + 1 q ( ς ) : = ς p + ∑ j = p + 1 ∞ [ μ + 1 ] q , j − p [ j − 1 ] q ! ς j , ς ∈ Δ .$
A simple computation shows that
$D p , λ μ , q F ( ς ) : = ς p + ∑ j = p + 1 ∞ [ j ] q ! λ j − p j − p − 1 e − λ j − p p q [ μ + 1 ] q , j − p j − p ! a j ς j = z p + ∑ j = p + 1 ∞ ϕ j a j ς j ( 0 < λ ≤ 1 , μ > p , 0 < q < 1 , ς ∈ Δ ) .$
where
$ϕ j = [ j ] q ! λ j − p j − p − 1 e − λ j − p p q [ μ + 1 ] q , j − p j − p ! .$
For $δ ≥ 0 ,$ with the aid of the operator $D p , λ μ , q$ one can defined the linear q-differential Borel operator $A p → A p$ as follows:
$G p , q , λ , δ μ , 0 F ( ς ) : = D p , λ μ , q F ( ς ) G p , q , λ , δ μ , 1 F ( ς ) : = 1 − δ G p , q , λ , δ μ , 0 F ( ς ) + δ ς p G p , q , λ , δ μ , 0 F ( ς ) ′ = ς p + ∑ j = p + 1 ∞ [ j ] q ! λ j − p j − p − 1 e − λ j − p p q [ μ + 1 ] q , j − p j − p ! 1 + δ j p − 1 a j ς j G p , q , λ , δ μ , 2 F ( ς ) : = 1 − δ G p , q , λ , δ μ , 1 F ( ς ) + δ ς p G p , q , λ , δ μ , 1 F ( ς ) ′ = ς p + ∑ j = p + 1 ∞ [ j ] q ! λ j − p j − p − 1 e − λ j − p p q [ μ + 1 ] q , j − p j − p ! 1 + δ j p − 1 2 a j ς j . . . G p , q , λ , δ μ , m F ( ς ) : = ς p + ∑ j = p + 1 ∞ [ j ] q ! λ j − p j − p − 1 e − λ j − p p q [ μ + 1 ] q , j − p j − p ! 1 + δ j p − 1 m a j ς j , m ∈ N 0 = N ∪ 0 , δ ≥ 0 , 0 < λ ≤ 1 , μ > p , 0 < q < 1 .$
From the relation (10), we can easily deduce that the next relations held for all $F ∈ A p$:
$( i ) ς G p , q , λ , δ μ , m F ( ς ) ′ = μ G p , q , λ , δ μ − 1 , m F ( ς ) − μ − p G p , q , λ , δ μ , m F ( ς ) ,$
and
$( ii ) δ ς G p , q , λ , δ μ , m F ( ς ) ′ = p G p , q , λ , δ μ , m + 1 F ( ς ) − p 1 − δ G p , q , λ , δ μ , m F ( ς )$
Remark 1.
By particularizing the parameters $p$ and $m ,$ we derive the following operators based on Borel distribution:
(1)
Letting $p = 1 ,$ we obtain that $G 1 , q , λ , δ μ , m = : I q , λ , δ μ , m$, where the operator $I q , λ , δ μ , m$ is defined as follows:
$I q , λ , δ μ , m F ( ς ) : = ς + ∑ j = 2 ∞ [ j ] q ! λ j − 1 j − 2 e − λ j − 1 [ μ + 1 ] q , j − 1 j − 1 ! 1 + δ j − 1 m a j ς j ;$
(2)
Letting $p = 1$ and $m = 0 ,$ we deduce that $G 1 , q , λ , δ μ , 0 = : B λ μ , q$, where the operator $B λ μ , q$, introduced by El-Deeb and Murugusundaramoorthy [16];
(3)
Letting $q → 1 −$ and $p = 1$, we deduce that $lim q → 1 − G 1 , q , λ , δ μ , m : = R λ , δ μ , m ,$ where the operator $R λ , δ μ , m$ is defined as follows
$R λ , δ μ , m F ( ς ) : = ς + ∑ j = 2 ∞ j λ j − 1 j − 2 e − λ j − 1 ( μ + 1 ) j − 1 1 + δ j − 1 m a j ς j ;$
(4)
Putting $q → 1 − ,$$p = 1$ and $m = 0$, we obtain that $lim q → 1 − G 1 , q , λ , δ μ , 0 : = M λ μ ,$ where the operator $M λ μ$, studied byEl-Deeb and Murugusundaramoorthy [16].
Now we introduce the following classes $S p k α , C p k α$ and $K p k β , α$ of the class $A p$ for $0 ≤ α , β < p , p ∈ N$ and $k ≥ 2$ as follows:
$S p k α = F : F ∈ A p and ς F ′ ( ς ) F ( ς ) ∈ P p , k α , ς ∈ Δ ,$
$C p k α = F : F ∈ A p and 1 + ς F ′ ′ ( ς ) F ′ ( ς ) ∈ P p , k α , ς ∈ Δ ,$
and
$K p k β , α = F : F ∈ A p , g ∈ S p 2 α and ς F ′ ( ς ) g ( ς ) ∈ P p , k β , ς ∈ Δ .$
Obviously, we know that
$F ( ς ) ∈ C p k α ⇔ ς F ′ ( ς ) p ∈ S p k α .$
Remark 2.
By particularizing the parameter $k ,$ we obtain the following classes:
(i)
$S p 2 α = S p ∗ ( α )$$0 ≤ α < p , p ∈ N ,$ where $S p ∗ ( α )$ is the well-known class of $p −$valently starlike functions of order α and was studied by Patil and Thakare [17];
(ii)
$C p 2 α = C p α$$0 ≤ α < p , p ∈ N ,$ where $C p α$ is the well-known class of $p −$valently convex functions of order α and was studied by Owa [18];
(iii)
$K p 2 β , α = K p ( β , α )$$0 ≤ α < p , p ∈ N ,$ where $K p ( β , α )$ is the class of all $p −$valently close-to-convex functions of order β and type $α$ and was introduced by Aouf [19].
Next, by making use of the operator defined by (10), we obtain the following subclasses $S p , q , λ , δ μ , m , k α$, $C p , q , λ , δ μ , m , k α$ and $K p , q , λ , δ μ , m , k β , α$ of the class $A p$ as follows:
$S p , q , λ , δ μ , m , k α = F : F ∈ A p and G p , q , λ , δ μ , m F ( ς ) ∈ S p k α , ς ∈ Δ ,$
$C p , q , λ , δ μ , m , k α = F : F ∈ A p and G p , q , λ , δ μ , m F ( ς ) ∈ C p k α , ς ∈ Δ ,$
and
$K p , q , λ , δ μ , m , k β , α = F : F ∈ A p and G p , q , λ , δ μ , m F ( ς ) ∈ K p k β , α , ς ∈ Δ .$
We can easily see that
$F ( ς ) ∈ C p , q , λ , δ μ , m , k α ⇔ ς F ′ ( ς ) p ∈ S p , q , λ , δ μ , m , k α$
In order to establish our main results, we will require the following lemmas.
Lemma 1
([20,21]). Let $Φ ( r , s )$ be complex valued function, $Φ :$$D → C ,$$D ⊂ C × C$$( C$ is the complex plane) and let $r = r 1 + i r 2 ,$ $s = s 1 + i s 2 .$ Suppose that $Φ ( r , s )$ satisfies the following conditions:
(i)
$Φ ( r , s )$ is continuous in a domain $D ;$
(ii)
$( 1 , 0 ) ∈ D$ and $ℜ Φ ( 1 , 0 ) > 0 ;$
(iii)
$ℜ Φ ( i r 2 , s 1 ) ≤ 0$ for all $( i r 2 , s 1 ) ∈ D$ and such that $s 1 ≤ − 1 2 ( 1 + r 2 2 ) .$
Let $h ( ς ) = 1 + ∑ m = 1 ∞ c m ς m ,$ be regular in Δ such that $( h ( ς ) , ς h ′ ( ς ) ) ∈ D$ for all $ς ∈ Δ .$ If
$ℜ Φ ( h ( ς ) , ς h ′ ( ς ) ) > 0 ( ς ∈ Δ ) ,$
then
$ℜ h ( ς ) > 0 ( ς ∈ Δ ) .$
Lemma 2
([22]). Let Φ be convex and $F$ be starlike in Δ. Then, for Υ analytic in Δ with $Υ ( 0 ) = 1$, $Φ ∗ Υ F Φ ∗ F$ is contained in the convex hull of $Υ ( Δ )$.

## 3. Inclusion Properties Involving the Operator $G p , q , λ , δ μ , m$

Further, we assume throughout this paper that $k ≥ 2 , p ∈ N , m ∈ N 0 , δ ≥ 0 , 0 < λ ≤ 1 , 0 < q < 1 , ς ∈ Δ$ and the power are the principal values.
Theorem 1.
For $0 ≤ ζ ≤ α < p$ and $μ > p ,$ then
$S p , q , λ , δ μ − 1 , m , k α ⊂ S p , q , λ , δ μ , m , k ζ ,$
where ζ is given by
$ζ = 2 p − 2 α p − μ 2 μ − 2 p − 2 α + 1 2 + 8 p − 2 α p − μ + 2 μ − 2 p − 2 α + 1 .$
Proof.
Assume that $F ∈ S p , q , λ , δ μ − 1 , m , k α$ and let
$ς G p , q , λ , δ μ , m F ( ς ) ′ G p , q , λ , δ μ , m F ( ς ) = M ( ς ) = ( p − ζ ) h ( ς ) + ζ .$
where
$h ( ς ) = k 4 + 1 2 h 1 ( ς ) − k 4 − 1 2 h 2 ( ς )$
and $h i ( z )$ $i = 1 , 2$ are analytic in $Δ$ with $h i ( 0 ) = 1 , i = 1 , 2 .$ Using (11) and (19), we have
$μ G p , q , λ , δ μ − 1 , m F ( ς ) G p , q , λ , δ μ , m F ( ς ) = ( p − ζ ) h ( ς ) + ζ − μ + p .$
By computing the logarithmical derivative of (21) with respect to $ς$, we have
$ς G p , q , λ , δ μ − 1 , m F ( ς ) ′ G p , q , λ , δ μ − 1 , m F ( ς ) − α = ζ − α + ( p − ζ ) h ( ς ) + ( p − ζ ) ς h ′ ( ς ) ( p − ζ ) h ( ς ) + ζ − μ + p .$
Now we show that $M ( ς ) ∈ P p , k α$ or $h i ( ς ) ∈ P , i = 1 , 2 .$ From (20) and (22), we have
$ς G p , q , λ , δ μ − 1 , m F ( ς ) ′ G p , q , λ , δ μ − 1 , m F ( ς ) − α = k 4 + 1 2 ζ − α + ( p − ζ ) h 1 ( ς ) + ( p − ζ ) ς h 1 ′ ( ς ) ( p − ζ ) h 1 ( ς ) + ζ − μ + p$
$− k 4 − 1 2 ζ − α + ( p − ζ ) h 2 ( ς ) + ( p − ζ ) ς h 2 ′ ( ς ) ( p − ζ ) h 2 ( ς ) + ζ − μ + p$
and this implies that
$ℜ ζ − α + ( p − ζ ) h i ( ς ) + ( p − ζ ) ς h i ′ ( ς ) ( p − ζ ) h i ( ς ) + ζ − μ + p > 0 ς ∈ Δ ; i = 1 , 2 .$
We form the function $Φ ( r , s )$ by choosing $r = h i ( ς )$ and $s = ς h i ′ ( ς ) .$ Thus
$Φ ( r , s ) = ζ − α + ( p − ζ ) r + ( p − ζ ) s ( p − ζ ) r + ζ − μ + p .$
Then, we have
(i)
$Φ ( r , s )$ is continuous function in $D = C \ ζ − μ + p ζ − p × C ;$
(ii)
$( 1 , 0 ) ∈ D$ and $ℜ Φ ( 1 , 0 ) = p − α > 0 ;$
(iii)
$ℜ Φ ( i r 2 , s 1 ) = ℜ ζ − α + ( p − ζ ) i r 2 + ( p − ζ ) s 1 ( p − ζ ) i r 2 + ζ − μ + p = ζ − α + ( p − ζ ) ζ − μ + p s 1 ( p − ζ ) 2 r 2 2 + ζ − μ + p 2 ≤ ζ − α − ( p − ζ ) ζ − μ + p ( 1 + r 2 2 ) 2 ( p − ζ ) 2 r 2 2 + ζ − μ + p 2 = R + E r 2 2 2 C ,$
for all $( i r 2 , s 1 ) ∈ D$ such that $s 1 ≤ − 1 2 ( 1 + r 2 2 ) ,$
where
$R = 2 ζ − α ζ − μ + p 2 − ( p − ζ ) ζ − μ + p ,$
$E = 2 ζ − α ( p − ζ ) 2 − ( p − ζ ) ζ − μ + p ,$
$C = ( p − ζ ) 2 r 2 2 + ζ − μ + p 2 .$
We note that $ℜ Φ ( i r 2 , s 1 ) < 0$, if and only if $R ≤ 0 , E < 0$ and $C > 0 .$ From $R ≤ 0$, we obtain $ζ$ as given by (18), and from $0 ≤ ζ < α < p ,$ we have $E < 0 .$ By applying Lemma 1, $h i ( ς ) ∈ P i = 1 , 2$ and consequently $M ( ς ) ∈ P p , k γ$ for $ς ∈ Δ .$ This completes the proof of Theorem 1. □
Theorem 2.
For $0 ≤ ζ ≤ α < p$ and $μ > p ,$ then
$C p , q , λ , δ μ − 1 , m , k α ⊂ C p , q , λ , δ μ , m , k ζ ,$
where ζ is given by (18).
Proof.
Let
$F ∈ C p , q , λ , δ μ − 1 , m , k α ⇒ G p , q , λ , δ μ − 1 , m F ( ς ) ∈ C p k α ⇒ ς G p , q , λ , δ μ − 1 , m F ( ς ) ′ p ∈ S p k α ⇒ G p , q , λ , δ μ − 1 , m ς F ′ ( ς ) p ∈ S p k α ⇒ ς F ′ ( ς ) p ∈ S p , q , λ , δ μ − 1 , m , k α ⊂ S p , q , λ , δ μ , m , k ζ ⇒ G p , q , λ , δ μ , m ς F ′ ( ς ) p ∈ S p k ζ ⇒ G p , q , λ , δ μ , m F ( ς ) ∈ C p k ζ ⇒ F ∈ C p , q , λ , δ μ , m , k ζ .$
This completes the proof of Theorem 2. □
Theorem 3.
For $0 ≤ β ≤ α < p$ and $μ > p ,$ then
$K p , q , λ , δ μ − 1 , m , k β , α ⊂ K p , q , λ , δ μ , m , k β , α .$
Proof.
Let $F ∈ K p , q , λ , δ μ − 1 , m , k β , α .$ Then, there exists $G ( ς ) ∈ S p 2 α ≡ S p ∗ α$ such that
$ς G p , q , λ , δ μ − 1 , m F ( ς ) ′ G ( ς ) ∈ P p , k β .$
Then
$G ( ς ) = G p , q , λ , δ μ − 1 , m g ( ς ) ∈ S p , q , λ , δ μ − 1 , m , 2 α .$
We set
$ς G p , q , λ , δ μ , m F ( ς ) ′ G p , q , λ , δ μ , m g ( ς ) = R ( ς ) = p − β h ( ς ) + β ,$
where $h ( ς )$ is given by (20). By using (11) in (23), we get
$ς G p , q , λ , δ μ − 1 , m F ( ς ) ′ G p , q , λ , δ μ − 1 , m g ( ς ) = ς G p , q , λ , δ μ , m ς F ′ ( ς ) ′ + μ − p G p , q , λ , δ μ , m ς F ′ ( ς ) ς G p , q , λ , δ μ , m g ( ς ) ′ + μ − p G p , q , λ , δ μ , m g ( ς ) .$
Furthermore, $G ( ς ) ∈ S p , q , λ , δ μ − 1 , m , 2 α$ and by using Theorem 1, with $k = 2 ,$ we have $G ( ς ) ∈ S p , q , λ , δ μ , m , 2 α .$ Therefore, we can write
$ς G p , q , λ , δ μ , m g ( ς ) ′ G p , q , λ , δ μ , m g ( ς ) = R 0 ( ς ) = p − α q ( ς ) + α q ∈ P k ,$
where $q ( ς ) = 1 + c 1 ς + c 2 ς 2 + . . .$ is analytic and $q ( 0 ) = 1$ in $Δ$. By differentiating (24) with respect to $ς$, we have
$ς G p , q , λ , δ μ , m ς F ′ ( ς ) ′ = ς G p , q , λ , δ μ , m g ( ς ) ′ R ( ς ) + ς R ′ ( ς ) G p , q , λ , δ μ , m g ( ς )$
then
$ς G p , q , λ , δ μ , m ς f ′ ( ς ) ′ G p , q , λ , δ μ , m g ( ς ) = ς R ′ ( ς ) + R 0 ( ς ) R ( ς ) .$
From (25) and (27), we obtain
$ς G p , q , λ , δ μ − 1 , m F ( ς ) ′ G p , q , λ , δ μ − 1 , m g ( ς ) = ς R ′ ( ς ) + R 0 ( ς ) R ( ς ) + μ − p R ( ς ) R 0 ( ς ) + μ − p$
so that
$ς G p , q , λ , δ μ − 1 , m f ( ς ) ′ G p , q , λ , δ μ − 1 , m g ( ς ) = R ( ς ) + ς R ′ ( ς ) R 0 ( ς ) + μ − p .$
Let
$R ( ς ) = k 4 + 1 2 p − β h 1 ( ς ) + β − k 4 − 1 2 p − β h 2 ( ς ) + β$
and
$R 0 ( ς ) + μ − p = p − α q ( ς ) + α + μ − p .$
We intend to show that $R ∈ P p , k β$ or $h i ∈ P$ for $i = 1 , 2 .$ Then, we can say that $ℜ R 0 ( ς ) + μ − p > 0 .$ From (24) and (28), we have
$ς G p , q , λ , δ μ − 1 , m F ( ς ) ′ G p , q , λ , δ μ − 1 , m g ( ς ) − β = k 4 + 1 2 p − β h 1 ( ς ) + p − β ς h 1 ′ ( ς ) p − α q ( ς ) + α + μ − p$
$− k 4 − 1 2 p − β h 2 ( ς ) + p − β ς h 2 ′ ( ς ) p − α q ( ς ) + α + μ − p$
and this implies that
$ℜ p − β h i ( ς ) + p − β ς h i ′ ( ς ) p − α q ( ς ) + α + μ − p > 0 ς ∈ Δ , i = 1 , 2 .$
We form the function $Φ ( r , s )$ by choosing $r = h i ( ς )$ and $s = ς h i ′ ( ς ) .$ Thus,
$Φ ( r , s ) = ( p − β ) r + ( p − β ) s p − α q ( ς ) + α + μ − p .$
Then
(i)
$Φ ( r , s )$ is continuous in $D = C × C ;$
(ii)
$( 1 , 0 ) ∈ D$ and $ℜ Φ ( 1 , 0 ) = p − β > 0 ;$
(iii)
$ℜ Φ ( i r 2 , s 1 ) = ℜ ( p − β ) i u 2 + ( p − β ) v 1 ( p − α ) q 1 + i q 2 + α + μ − p = ( p − β ) ( p − α ) q 1 + α + μ − p s 1 ( p − α ) q 1 + α + μ − p 2 + ( p − α ) 2 q 2 2 ≤ − ( p − β ) ( p − α ) q 1 + α + μ − p ( 1 + r 2 2 ) 2 ( p − α ) q 1 + α + μ − p 2 + ( p − α ) 2 q 2 2 < 0 ,$
for all $( i r 2 , s 1 ) ∈ D$ such that $s 1 ≤ − 1 2 ( 1 + r 2 2 ) .$
Byapplying Lemma 1, we have $ℜ h i ( ς ) > 0$ for $i = 1 , 2$ and consequently $R ( ς ) ∈ P p , k β$ for $ς ∈ Δ .$ This completes the proof of Theorem 3. □

## 4. Inclusion Properties Involving the Integral Operator $J δ , p$

The generalized Bernardi operator is defined by (see [23])
$J δ , p ( F ) ( ς ) = δ + p ς δ ∫ 0 ς t δ − 1 F ( t ) d t δ > − p ,$
which satisfies the following relationship:
$ς J δ , p ( F ) ( ς ) ′ = δ + p F ( ς ) − δ J δ , p ( F ) ( ς ) .$
Theorem 4.
If $0 ≤ α < p ,$$k ≥ 2$ and $F ∈ S p , q , λ , δ μ , m , k α ,$ then $J δ , p ( F ) ∈ S p , q , λ , δ μ , m , k α δ ≥ 0 .$
Proof.
Let
$ς G p , q , λ , δ μ , m J δ , p ( F ) ( ς ) ′ G p , q , λ , δ μ , m J δ , p ( F ) ( ς ) = R ( ς ) = ( p − α ) h ( ς ) + α ,$
where $h ( ς )$, given by (20). Using (31), we have
$ς G p , q , λ , δ μ , m J δ , p ( F ) ( ς ) ′ = δ + p G p , q , λ , δ μ , m F ( ς ) − δ G p , q , λ , δ μ , m J δ , p ( F ) ( ς ) .$
From (32) and (33), we have
$δ + p G p , q , λ , δ μ , m F ( ς ) G p , q , λ , δ μ , m J δ , p ( F ) ( ς ) = ( p − α ) h ( ς ) + α + δ .$
By computing the logarithmical derivative of (34) with respect to $ς$ and multiplying by $ς$, we have
$ς G p , q , λ , δ μ , m F ( ς ) ′ G p , q , λ , δ μ , m F ( ς ) − α = ( p − α ) h ( ς ) + ( p − α ) ς h ′ ( ς ) ( p − α ) h ( ς ) + α + δ .$
Now, we show that $R ( ς ) ∈ P p , k α$ or $h i ∈ P$ for $i = 1 , 2 .$ From (20) and (35), we have
$ς G p , q , λ , δ μ , m F ( ς ) ′ G p , q , λ , δ μ , m F ( ς ) − α = k 4 + 1 2 ( p − α ) h 1 ( ς ) + ( p − α ) ς h 1 ′ ( ς ) ( p − α ) h 1 ( ς ) + α + δ$
$− k 4 − 1 2 ( p − α ) h 2 ( ς ) + ( p − α ) ς h 2 ′ ( ς ) ( p − α ) h 2 ( ς ) + α + δ$
and this implies that
$ℜ ( p − α ) h i ( ς ) + ( p − α ) ς h i ′ ( ς ) ( p − α ) h i ( ς ) + α + δ > 0 ς ∈ Δ ; i = 1 , 2 .$
We form the function $Φ ( r , s )$ by choosing $r = h i ( ς )$ and $s = ς h i ′ ( ς ) .$ Thus
$Φ ( r , s ) = ( p − α ) r + ( p − α ) s ( p − α ) r + α + δ .$
Clearly, conditions (i), (ii) and (iii) of Lemma 1 are satisfied. Byapplying Lemma 1, we have $ℜ h i ( ς ) > 0$ for $i = 1 , 2$ and consequently $J δ , p ( F ) ∈ S p , q , λ , δ μ , m , k α$ for $ς ∈ Δ .$ This completes the proof of Theorem 1. □
Theorem 5.
If $0 ≤ α < p ,$$k ≥ 2$ and $F ∈ C p , q , λ , δ μ , m , k α ,$ then $J δ , p ( F ) ∈ C p , q , λ , δ μ , m , k α δ ≥ 0 .$
Proof.
Let
$F ∈ C p , q , λ , δ μ , m , k α ⇔ ς F ′ ( ς ) p ∈ S p , q , λ , δ μ , m , k α .$
By applying Theorem 4, we have
$J δ , p ς F ′ ( ς ) p ∈ S p , q , λ , δ μ , m , k α ⇔ ς J δ , p ( F ) ( ς ) ′ p ∈ S p , q , λ , δ μ , m , k α ⇔ J δ , p ( F ) ( ς ) ∈ C p , q , λ , δ μ , m , k α ,$
which evidently proves Theorem 5. □

## 5. Inclusion Properties by Convolution

Theorem 6.
Let Φ be a convex function and $F ∈ S p , q , λ , δ μ , m , 2 p γ ,$ then $G ∈ S p , q , λ , δ μ , m , 2 p γ ,$ where $G = F ∗ Φ$ and $0 ≤ γ < 1 .$
Proof.
To show that $G = F ∗ Φ ∈ S p , q , λ , δ μ , m , 2 p γ 0 ≤ γ < 1 ,$ it sufficient to show that $ς G p , q , λ , δ μ , m G ′ p G p , q , λ , δ μ , m G$ contained in the convex hull of $Υ Δ .$ Now
$ς G p , q , λ , δ μ , m G ′ p G p , q , λ , δ μ , m G = Φ ∗ Υ G p , q , λ , δ μ , m F Φ ∗ G p , q , λ , δ μ , m F ,$
where $Υ = ς G p , q , λ , δ μ , m F ′ p G p , q , λ , δ μ , m F$ is analytic in $Δ$ and $Υ ( 0 ) = 1 .$ From Lemma 2, we can see that $ς G p , q , λ , δ μ , m G ′ p G p , q , λ , δ μ , m G$ is contained in the convex hull of $Υ Δ ,$ since $ς G p , q , λ , δ μ , m G ′ p G p , q , λ , δ μ , m G$ is analytic in $Δ$ and
$Υ Δ ⊆ Ω = w : ς G p , q , λ , δ μ , m w ( ς ) ′ p G p , q , λ , δ μ , m w ( ς ) ∈ P ( γ ) ,$
then $ς G p , q , λ , δ μ , m G ′ p G p , q , λ , δ μ , m G$ lies in $Ω ,$ this implies that $G = F ∗ Φ ∈ S p , q , λ , δ μ , m , 2 p γ$. □
Theorem 7.
Let Φ be a convex function and $F ∈ C p , q , λ , δ μ , m , 2 p γ ,$ then $G ∈ C p , q , λ , δ μ , m , 2 p γ ,$ where $G = F ∗ Φ$ and $0 ≤ γ < 1 .$
Proof.
Let $F ∈ C p , q , λ , δ μ , m , 2 p γ ,$ then, by using (13), we have
$ς F ′ ( ς ) p ∈ S p , q , λ , δ μ , m , 2 p γ$
and hence by using Theorem 6, we get
$ς F ′ ( ς ) p ∗ Φ ( ς ) ∈ S p , q , λ , δ μ , m , 2 p γ ⇒ ς F ∗ Φ ′ ( ς ) p ∈ S p , q , λ , δ μ , m , 2 p γ .$
Now applying (13) again, we obtain $G = F ∗ Φ ∈ C p , q , λ , δ μ , m , 2 p γ$, which evidently proves Theorem 7. □
Remark 3.
Particularizing the parameters $q$ and $m$ in the results of this paper, we derive various results for different operators.

## 6. Conclusions

In the present survey, we propose new subclasses of p-valent functions by making use of the linear q-differential Borel operator. The applications of this interesting operator are discussed. Inclusion properties and certain integral preserving relations were aimed to be our main concern.

## Author Contributions

Conceptualization, A.C. and S.M.E.-D.; methodology, S.M.E.-D.; validation, A.C., E.-R.B. and S.M.E.-D.; formal analysis, E.-R.B.; investigation, A.C., E.-R.B. and S.M.E.-D.; writing—original draft, S.M.E.-D.; writing—review & editing, A.C. and S.M.E.-D.; visualization, E.-R.B.; supervision, S.M.E.-D.; project administration, S.M.E.-D. All authors have read and agreed to the published version of the manuscript.

## Funding

The research was funded by the University of Oradea, Romania.

Not applicable.

Not applicable.

## Data Availability Statement

No data were used to support this study.

## Conflicts of Interest

The authors declare no conflict of interest.

## References

1. Aouf, M.K. A generalization of functions with real part bounded in the mean on the unit disc. Math. Japon. 1988, 33, 175–182. [Google Scholar]
2. Padmanabhan, K.S.; Parvatham, R. Properties of a class of functions with bounded boundary rotation. Ann. Polon. Math. 1975, 31, 311–323. [Google Scholar] [CrossRef] [Green Version]
3. Pinchuk, B. Functions with bounded boundary rotation. Isr. Math. 1971, 10, 7–16. [Google Scholar] [CrossRef]
4. Robertson, M.S. Variational formulas for several classes of analytic functions. Math. Z. 1976, 118, 311–319. [Google Scholar] [CrossRef]
5. Wanas, A.K.; Khuttar, J.A. Applications of Borel distribution series on analytic functions. Earthline J. Math. Sci. 2020, 4, 71–82. [Google Scholar] [CrossRef]
6. El-Deeb, S.M.; Murugusundaramoorthy, G.; Alburaikan, A. Bi-Bazilevic functions based on the Mittag–Leffler-type Borel distribution associated with Legendre polynomials. J. Math. Comput. Sci. 2022, 24, 173–183. [Google Scholar] [CrossRef]
7. Murugusundaramoorthy, G.; El-Deeb, S.M. Second Hankel determinant for a class of analytic functions of the Mittag–Leffler-type Borel distribution related with Legendre polynomials. Twms J. Appl. Eng. Math. 2022, 12, 1247–1258. [Google Scholar]
8. Srivastava, H.M. Operators of basic (or q-) calculus and fractional q-calculus and their applications in Geometric Function theory of Complex Analysis. Iran. J. Sci. Technol. Trans. Sci. 2020, 44, 327–344. [Google Scholar] [CrossRef]
9. Gasper, G.; Rahman, M. Basic hypergeometric series (with a Foreword by Richard Askey). In Encyclopedia of Mathematics and Its Applications; Cambridge University Press: Cambridge, UK, 1990; Volume 35. [Google Scholar]
10. Jackson, F.H. On q-definite integrals. Quart. J. Pure Appl. Math. 1910, 41, 193–203. [Google Scholar]
11. Al-Shbeil, I.; Shaba, T.G.; Cătaş, A. Second Hankel Determinant for the Subclass of Bi-Univalent Functions Using q-Chebyshev Polynomial and Hohlov Operator. Fractal Fract. 2022, 6, 186. [Google Scholar] [CrossRef]
12. Abu Risha, M.H.; Annaby, M.H.; Ismail, M.E.H.; Mansour, Z.S. Linear q-difference equations. Z. Anal. Anwend. 2007, 26, 481–494. [Google Scholar] [CrossRef] [Green Version]
13. Cătaş, A. On the Fekete-Szegö problem for certain classes of meromorphic functions using p,q-derivative operator and a p,q-wright type hypergeometric function. Symmetry 2021, 13, 2143. [Google Scholar] [CrossRef]
14. El-Deeb, S.M.; El-Matary, B.M. Coefficient boundeds of p-valent function connected with q-analogue of Salagean operator. Appl. Math. Inf. Sci. 2020, 14, 1057–1065. [Google Scholar]
15. Jackson, F.H. On q-functions and a certain difference operator. Trans. R. Soc. Edinb. 1909, 46, 253–281. [Google Scholar] [CrossRef]
16. El-Deeb, S.M.; Murugusundaramoorthy, G. Applications on a subclass of β-uniformly starlike functions connected with q-Borel distribution. Asian-Eur. J. Math. 2022, 15, 1–20. [Google Scholar] [CrossRef]
17. Patil, D.A.; Thakare, N.K. On convex hulls and extreme points of p-valent starlike and convex classes with applications. Bull. Math. Soc. Sci. Math. R. S. Roum. 1983, 27, 145–160. [Google Scholar]
18. Owa, S. On certain classes of p-valent functions with negative coefficient. Simon Stevin 1985, 59, 385–402. [Google Scholar]
19. Aouf, M.K. On a class of p-valent close-to-convex functions, Internat. J. Math. Math. Sci. 1988, 11, 259–266. [Google Scholar] [CrossRef] [Green Version]
20. Miller, S.S. Differential inequalities and Caratheodory function. Bull. Am. Math. Soc. 1975, 8, 79–81. [Google Scholar] [CrossRef] [Green Version]
21. Miller, S.S.; Mocanu, P.T. Second order differential inequalities in the complex plane. J. Math. Anal. Appl. 1978, 65, 289–305. [Google Scholar] [CrossRef] [Green Version]
22. Ruscheweyh, S.; Shiel-Small, T. Hadmard product of schlicht functions and Polya-Schoenberg conjecture. Comment. Math. Helv. 1973, 48, 119–135. [Google Scholar] [CrossRef]
23. Choi, J.H.; Saigo, M.; Srivastava, H.M. Some inclusion properties of a certain family of integral operators. J. Math. Anal. Appl. 2002, 276, 432–445. [Google Scholar] [CrossRef]
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Cătaş, A.; Borşa, E.-R.; El-Deeb, S.M. Subclasses of p-Valent Functions Associated with Linear q-Differential Borel Operator. Mathematics 2023, 11, 1742. https://doi.org/10.3390/math11071742

AMA Style

Cătaş A, Borşa E-R, El-Deeb SM. Subclasses of p-Valent Functions Associated with Linear q-Differential Borel Operator. Mathematics. 2023; 11(7):1742. https://doi.org/10.3390/math11071742

Chicago/Turabian Style

Cătaş, Adriana, Emilia-Rodica Borşa, and Sheza M. El-Deeb. 2023. "Subclasses of p-Valent Functions Associated with Linear q-Differential Borel Operator" Mathematics 11, no. 7: 1742. https://doi.org/10.3390/math11071742

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