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Article

A New Pseudo-Type κ-Fold Symmetric Bi-Univalent Function Class

by
Sondekola Rudra Swamy
1 and
Luminita-Ioana Cotîrlă
2,*
1
Department of Information Science and Engineering, Acharya Institute of Technology, Bengaluru 560 107, Karnataka, India
2
Department of Mathematics, Tehnical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
*
Author to whom correspondence should be addressed.
Axioms 2023, 12(10), 953; https://doi.org/10.3390/axioms12100953
Submission received: 25 August 2023 / Revised: 7 October 2023 / Accepted: 7 October 2023 / Published: 10 October 2023
(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

Abstract

:
We introduce and study a new pseudo-type $κ$-fold symmetric bi-univalent function class that meets certain subordination conditions in this article. For functions in the newly formed class, the initial coefficient bounds are obtained. For members in this class, the Fekete–Szegö issue is also estimated. In addition, we uncover pertinent links to previous results and give a few observations.
MSC:
30C45; 30C50

1. Preliminaries

Let ${ ς ∈ C : | ς | < 1 }$ = $D$, where $C$ is the set of all complex numbers. Let $A$ denote the class of all regular functions of the type
$s ( ς ) = ς + ∑ j = 2 ∞ d j ς j$
with $s ( 0 ) = s ′ ( 0 ) − 1 = 0$, $ς ∈ D$ and $S$ denote the subfamily of functions $∈ A$ which are univalent in $D$. For $τ ≥ 1$, the class of $τ$-pseudo-convex functions is defined as
$K τ = s ∈ A : R { ( ς s ′ ( ς ) ) ′ } τ s ( ς ) > 0 , ς ∈ D ,$
the class of $τ$-pseudo-starlike functions is given by
$S τ = s ∈ A : R ς { s ′ ( ς ) } τ s ( ς ) > 0 , ς ∈ D$
and the class of $τ$-pseudo-bounded turning is introduced as
$R τ = s ∈ A : R ( s ′ ( ς ) ) τ > 0 , , ( ς ∈ D ) ,$
The class $K τ$ was explored by Guney and Murugusundaramoorthy [1] and the class $S τ$ was examined in [2]. We note that $S 1 = S$. Al-Amiri and Reade [3] presented the class $M ( ν ) ( ν < 1 )$ of functions $s ∈ A$ with $s ′ ( ς ) ≠ 0 i n D$ which satisfy
$R ν ( ς s ′ ( ς ) ) ′ s ′ ( ς ) + ( 1 − ν ) s ′ ( ς ) > 0 , ( ς ∈ D ) .$
In [4], Sukhjit Singh and Sushma Gupta gave certain criteria for univalence by proving $ℜ ( s ′ ( ς ) ) > 0$, whenever
$R ν ( ς s ′ ( ς ) ) ′ s ′ ( ς ) + ( 1 − ν ) s ′ ( ς ) > ξ , ( 0 ≤ ν < 1 , 0 ≤ ξ < 1 , ς ∈ D ) .$
The Koebe theorem (see [5]) ensures that $s ( D )$, $s ∈ S$, contains a disc of radius 1/4. Thus, any function s admits an inverse $g = s − 1$ defined by $g ( s ( ς ) ) = ς$, and $s ( g ( ϰ ) ) = ϰ , | ϰ | < r 0 ( s ) , r 0 ( s ) ≥ 1 / 4$, $ς ∈ D$, $ϰ ∈ D ,$ where
$g ( ϰ ) = ϰ − d 2 ϰ 2 + ( 2 d 2 2 − d 3 ) ϰ 3 − ( 5 d 2 3 − 5 d 2 d 3 + d 4 ) ϰ 4 + ⋯$
If s$∈ S$ and $s − 1$$∈ S$, then a member s of $A$ given by (1) is called bi-univalent in $D$ and the collection of such functions in $D$ is symbolized by $σ$. For a brief study, and to know some interesting properties of the family $σ$, see [6]. Some subfamilies of the family $σ$ that are comparable to the well-known subfamilies of the family S have been introduced by Tan [7], Brannan and Taha [8], and Srivastava et al. [9]. In fact, as sequels to the above subfamilies of $σ$, a number of different subfamilies of $σ$ have since then been explored by many authors (see, for example, [10,11,12,13,14]). Most of these works are devoted to the study of the Fekete–Szegö issue of functions in various subfamilies of $σ$.
Let $N = { 1 , 2 , 3 , ⋯ }$ and $R = ( − ∞ , + ∞ )$.
If, for $κ ∈ N$, $s ( e 2 π i κ ς ) = e 2 π i κ s ( ς )$, $ς ∈ D$, then a regular function s is called a $κ$-fold symmetric ($κ$-FS). The function s, defined by $s ( ς ) = ( f ( ς κ ) ) 1 / κ$, $κ ∈ N$, $f ∈ S$, is univalent and maps $D$ into a $κ$-fold symmetry region. We indicate by $S κ$ the class of $κ$-fold symmetric univalent ($κ$-FSU) functions in $D$. A function $s ∈ S κ$ has the following form:
$s ( ς ) = ς + ∑ j = 1 ∞ d κ j + 1 · ς κ j + 1 ( κ ∈ N ; ς ∈ D ) .$
Clearly $S 1 = S$.
Similar to the idea of $S κ$, Srivastava et al. [15] investigated the class $σ κ$ of $κ$-fold symmetric bi-univalent ($κ$-FSBU) functions. A few intriguing findings were made, including the series
$s − 1 ( ϰ ) = ϰ − d κ + 1 ϰ κ + 1 + [ ( 1 + κ ) d κ + 1 2 − d 2 κ + 1 ] ϰ 2 κ + 1 − 1 2 ( 1 + κ ) ( 2 + 3 κ ) d κ + 1 3 − ( 2 + 3 κ ) d κ + 1 d 2 κ + 1 + d 3 κ + 1 ϰ 3 κ + 1 + ⋯ .$
when $s ∈ σ κ$.
Note that the functions
$s 1 ( ς ) = 1 2 l o g 1 + ς κ 1 − ς κ 1 / κ , s 2 ( ς ) = ς κ 1 − ς κ 1 / κ , s 3 ( ς ) = ( − l o g ( 1 − ς κ ) ) 1 / κ , ⋯ .$
with the corresponding inverses
$g 1 ( ϰ ) = e 2 ϰ κ − 1 e 2 ϰ κ − 1 1 / κ , g 2 ( ϰ ) = ϰ κ 1 + ϰ κ 1 / κ , g 3 ( ϰ ) = e ϰ κ − 1 e ϰ κ 1 / m , ⋯ .$
are elements of $σ κ$. We obtain (2) from (4) on taking $κ = 1$.
The focus on the initial coefficients of functions in some subfamilies of $σ κ$ is an interesting topic and this opened an area for many developments. New subfamilies of $σ κ$ were introduced and examined in depth by many researchers (see, for example, [16,17,18,19]). We mention here some recent works on this topic. Initial coefficient bounds for new subfamilies of $σ κ$ were determined in [20]. The Fekete–Szegö (FS) issue $| d 2 m + 1 − δ d m + 1 2 |$, $δ ∈ R$ (see [21]) for certain special families of $σ κ$ was examined by Swamy et al. [22,23]; and another special family of $σ κ$ satisfying certain subordination conditions was examined by Aldawish et al. [24]; initial coefficients estimates for elements belonging to certain new families of $σ κ$ were obtained by Breaz and Cotîrlă in [25] (see [26,27,28]), indicating the developments in this domain.
For functions $s 1$ and $s 2$ regular in $D$, $s 1$ is said to subordinate $s 2$, if there is a Schwarz function $ψ$ in $D$, such that $ψ ( 0 ) = 0$, $| ψ ( z ) | < 1$ and $s 1 ( z ) = s 2 ( ψ ( z ) ) , z ∈ D$. This subordination is indicated as $s 1 ≺ s 2$. If $s 2 ∈ S$, then $s 1 ( z ) ≺ s 2 ( z )$ is equivalent to $s 1 ( 0 ) = s 2 ( 0 )$ and $s 1 ( D ) ⊂ s 2 ( D )$.
Inspired by the efforts of Al-Amiri [3] and the authors of [19], we introduce a new class $P σ κ τ ( η , ν , φ )$, $η ∈ C ∗ = C − { 0 } , 0 ≤ ν ≤ 1 ,$ and $φ ( ς )$ is a regular function, such that $R ( φ ( ς ) ) > 0$, $φ ′ ( 0 ) > 0$, $φ ( 0 ) = 1$, $φ ( D )$ is symmetric with respect to the real axis. In Section 2, we estimate the upper bounds of $| d κ + 1 |$, $| d 2 κ + 1 |$ and $| d 2 κ + 1 − δ d κ + 1 2 |$ $( δ ∈ R )$, for functions that belong to the class $P σ κ τ ( η , ν , φ )$. We consider two special cases $Q σ κ ϱ ( η , ν , τ ) = P σ κ τ ( η , ν , 1 + ς 1 − ς ϱ ) , 0 < ϱ ≤ 1$ and $X σ κ ξ ( η , ν , τ ) = P σ κ τ ( η , ν , 1 + ( 1 − 2 ξ ) ς 1 − ς ) ,$$0 ≤ ξ < 1$, in Section 3 and Section 4, respectively. We also identify connections to existing results and present a few new observations.

2. The Class $P σ κ τ ( η , ν , φ )$

Throughout this paper, $s − 1 ( ϰ ) = g ( ϰ )$ is as in (4), $η ∈ C * = C ∖ { 0 }$, $ς ∈ D$, $ϰ ∈ D$ and $φ ( ς )$ will be a regular function such that $R ( φ ( ς ) ) > 0$, $φ ′ ( 0 ) > 0$, $φ ( 0 ) = 1$, and $φ ( D )$ is symmetric with respect to the real axis. An expansion of $φ ( ς )$ has the form:
$φ ( ς ) = 1 + B 1 ς + B 2 ς 2 + B 3 ς 3 + ⋯ ( B 1 > 0 ) .$
Let $P$ be the class of regular functions of the type $p ( ς ) = 1 + p 1 ς + p 2 ς 2 + p 3 ς 3 + ⋯ ,$ $R ( p ( ς ) ) > 0$. A $κ$-FS function $p κ$$∈ P$ is of the form $p κ ( ς ) = 1 + p κ ς κ + p 2 κ ς 2 κ + p 3 κ ς 3 κ + ⋯$ (see [29]).
Let $h ( ς )$ and $p ( ϰ )$ be regular in $D$ with $max { | h ( ς ) | , | p ( ϰ ) | }$< 1 and $h ( 0 ) = 0 = p ( 0 )$. We suppose that $h ( ς ) = h κ ς κ + h 2 κ ς 2 κ + h 3 κ ς 3 κ + ⋯$ and $p ( ϰ ) = p κ ϰ κ + p 2 κ ϰ 2 κ + p 3 κ ϰ 3 κ + ⋯$. Also, we assume that
$| h κ | < 1 ; | h 2 κ | ≤ 1 − | h κ | 2 ; | p κ | < 1 ; | p 2 κ | ≤ 1 − | p κ | 2 .$
After simple computations, using (5), we have
$φ ( h ( ς ) ) = 1 + B 1 h κ ς κ + ( B 1 h 2 κ + B 2 h κ 2 ) ς 2 κ + . . .$
and
$φ ( p ( ϰ ) ) = 1 + B 1 p κ ϰ κ + ( B 1 p 2 κ + B 2 p κ 2 ) ϰ 2 κ + . . . .$
Definition 1.
A function $s ∈ σ κ$ of the form (3) is said to be in the class $P σ κ τ ( η , ν , φ )$ if
$1 η ν { ( ς s ′ ( ς ) ) ′ } τ s ′ ( ς ) + ( 1 − ν ) ( s ′ ( ς ) ) τ − 1 + 1 ≺ φ ( ς )$
and
$1 η ν { ( ϰ g ′ ( ϰ ) ) ′ } τ g ′ ( ϰ ) + ( 1 − ν ) ( g ′ ( ϰ ) ) τ − 1 + 1 ≺ φ ( ϰ ) ,$
where $g = s − 1$, $τ ≥ 1$, $η ∈ C *$, and $0 ≤ ν < 1$.
Remark 1. (i) The subclass $P σ κ τ ( η , 0 , φ ) ≡ H σ κ τ ( η , φ )$, and was explored in [24].
(ii) $P σ κ 1 ( η , ν , φ ) ≡ I σ κ ( η , ν , φ )$ is the subclass of functions s $∈ σ κ$ satisfying
$1 η ν ( ς s ′ ( ς ) ) ′ s ′ ( ς ) + ( 1 − ν ) s ′ ( ς ) − 1 + 1 ≺ φ ( ς )$
and its inverse $g = s − 1$ satisfies
$1 η ν ( ϰ g ′ ( ϰ ) ) ′ g ′ ( ϰ ) + ( 1 − ν ) g ′ ( ϰ ) − 1 + 1 ≺ φ ( ϰ ) ,$
where $η ∈ C *$ and $0 ≤ ν < 1$.
Theorem 1.
If the function s given by (3) belongs to the family $P σ κ τ ( η , ν , φ )$ and $δ ∈ R$, then
$| d κ + 1 | ≤ | η | B 1 2 B 1 | { M ( 1 + κ ) + [ N τ ( τ − 1 ) + ( 1 − ( 1 + κ ) τ ) 2 ν ] ( 1 + κ ) 2 } η B 1 2 − 2 L 2 B 2 | + 2 L 2 B 1 ,$
$| d 2 κ + 1 | ≤ B 1 | η | M ; 0 < B 1 < 2 L 2 | η | M ( 1 + κ ) B 1 | η | M + 1 + κ 2 − L 2 | η | B 1 M 2 η 2 B 1 3 | { M ( 1 + κ ) + [ N τ ( τ − 1 ) + ( 1 − ( 1 + κ ) τ ) 2 ν ] ( 1 + κ ) 2 } η B 1 2 − 2 L 2 B 2 | + 2 L 2 B 1 ; B 1 ≥ 2 L 2 | η | M ( 1 + κ ) ,$
and
$| d 2 κ + 1 − δ d κ + 1 2 | ≤ B 1 | η | M ; | 1 + κ − 2 δ | < J | η | 2 B 1 3 | κ − 2 δ + 1 | | { M ( 1 + κ ) + [ N τ ( τ − 1 ) + ( 1 − ( 1 + κ ) τ ) 2 ν ] ( 1 + κ ) 2 } η B 1 2 − 2 L 2 B 2 | ; | 1 + κ − 2 δ | ≥ J ,$
where
$J = { M ( 1 + κ ) + [ N τ ( τ − 1 ) + 2 ν ( 1 − ( 1 + κ ) τ ) ] ( 1 + κ ) 2 } η B 1 2 − 2 L 2 B 2 η M B 1 2 .$
$L = ( 1 + κ ) ( τ ( 1 + ν κ ) − ν ) ,$
$M = ( τ ( 1 + 2 ν κ ) − ν ) ( 1 + 2 κ )$
and
$N = 1 + ν κ ( 2 + κ ) .$
Proof.
Let the function s of the form (3) belong to the family $P σ κ τ ( η , ν , φ )$. Then, we have regular functions $h , p : D ⟶ D$, $h ( 0 ) = p ( 0 ) = 0$ satisfying
$1 η ν { ( ς s ′ ( ς ) ) ′ } τ s ′ ( ς ) + ( 1 − ν ) ( s ′ ( ς ) ) τ − 1 + 1 = φ ( h ( ς ) ) ,$
and
$1 η ν { ( ϰ g ′ ( ϰ ) ) ′ } τ g ′ ( ϰ ) + ( 1 − ν ) ( g ′ ( ϰ ) ) τ − 1 + 1 = φ ( p ( ϰ ) ) .$
Using (3) in (16) and (17) we obtain:
$1 η ν [ { ( ς s ′ ( ς ) ) ′ } τ s ′ ( ς ) + ( 1 − ν ) ( s ′ ( ς ) ) τ − 1 + 1 =$
$1 η L d κ + 1 ς κ + M d 2 κ + 1 + ( 1 + κ ) 2 N τ ( τ − 1 ) 2 + ν ( 1 − ( 1 + κ ) τ ) d κ + 1 2 ς 2 κ + ⋯ + 1$
and
$1 η ν { ( ϰ g ′ ( ϰ ) ) ′ } τ g ′ ( ϰ ) + ( 1 − ν ) ( g ′ ( ϰ ) ) τ − 1 + 1 =$
$1 η − L d κ + 1 ϰ κ + M ( ( 1 + κ ) d κ + 1 2 − d 2 κ + 1 ) + ( 1 + κ ) 2 N τ ( τ − 1 ) 2 + ν ( 1 − ( 1 + κ ) τ ) d κ + 1 2 ϰ 2 κ + ⋯ + 1 ,$
where L, M, and N are as in (13), (14), and (15), respectively.
Comparing (7) and (18), we obtain
$L d κ + 1 = η B 1 h κ$
and
$M d 2 κ + 1 + N τ ( τ − 1 ) 2 + ν ( 1 − ( 1 + κ ) τ ) ( 1 + κ ) 2 d κ + 1 2 = η [ B 1 h 2 κ + B 2 h κ 2 ] .$
Comparing (8) and (19), we obtain
$− L d κ + 1 = η B 1 p κ$
and
$M ( ( κ + 1 ) d κ + 1 2 − d 2 κ + 1 ) + N τ ( τ − 1 ) 2 + ν ( 1 − ( 1 + κ ) τ ) ( 1 + κ ) 2 d κ + 1 2 = η [ B 1 p 2 κ + B 2 p κ 2 ] ,$
From (20) and (22), we obtain
$h κ = − p κ$
and
$2 L 2 d κ + 1 2 = η 2 B 1 2 ( h κ 2 + p κ 2 ) .$
We add (21) and (23) and then use (25) to obtain
$[ { M ( 1 + κ ) + [ N τ ( τ − 1 ) + ( 1 − ( 1 + κ ) τ ) 2 ν ] ( 1 + κ ) 2 } η B 1 2 − 2 L 2 B 2 ] d κ + 1 2 = η 2 B 1 3 h 2 κ + p 2 κ$
By using (6) and (20) in (26) for the coefficients $h 2 κ$ and $p 2 κ$, we obtain
$[ | { M ( 1 + κ ) + [ N τ ( τ − 1 ) + ( 1 − ( 1 + κ ) τ ) 2 ν ] ( 1 + κ ) 2 } η B 1 2 − 2 L 2 B 2 | + 2 L 2 B 1 ] | d κ + 1 | 2 ≤ 2 η 2 B 1 3 ,$
which implies (9).
We subtract (23) from (21) to find the bound on $| d 2 κ + 1 |$:
$d 2 κ + 1 = η B 1 ( h 2 κ − p 2 κ ) 2 M + 1 + κ 2 d κ + 1 2 .$
In view of (20), (24), (28) and applying (6), we obtain
$| d 2 κ + 1 | ≤ | η | B 1 M + 1 + κ 2 − L 2 | η | B 1 M ×$
$× 2 η 2 B 1 3 | { M ( 1 + κ ) + [ N τ ( τ − 1 ) + ( 1 − ( 1 + κ ) τ ) 2 ν ] ( 1 + κ ) 2 } η B 1 2 − 2 L 2 B 2 | + 2 L 2 B 1 ,$
which obtains (10), the desired assessment.
From (26) and (28), for $δ ∈ R$, we obtain
$d 2 κ + 1 − δ d κ + 1 2 = η B 1 2 Y ( δ ) + 1 M h 2 κ + Y ( δ ) − 1 M p 2 κ ,$
where
$Y ( δ ) = η B 1 2 ( κ − 2 δ + 1 ) { M ( 1 + κ ) + [ N τ ( τ − 1 ) + ( 1 − ( 1 + κ ) τ ) 2 ν ] ( κ + 1 ) 2 } η B 1 2 − 2 L 2 B 2 .$
In view of (6), we conclude that
$| d 2 κ + 1 − δ d κ + 1 2 | ≤ | η | B 1 M ; 0 ≤ | Y ( δ ) | < 1 M | η | B 1 | Y ( δ ) | ; | Y ( δ ) | ≥ 1 M ,$
form which we obtain (11) with J as in (12). So the proof is completed.
Remark 2.
We obtain Corollary 1 of [24] if $ν = 0$ in Theorem 1.
Choosing $τ = 1$ in $P σ κ τ ( η , ν , φ )$, we have the corollary given below:
Corollary 1.
Let $δ ∈ R$ and let the function s given by (3) be in the family $I σ κ ( η , ν , φ )$. Then,
$| d κ + 1 | ≤ | η | B 1 2 B 1 | { ( 1 + κ ) M 1 − 2 ν κ ( 1 + κ ) 2 } η B 1 2 − 2 L 1 2 B 2 | + 2 L 1 2 B 1 , | d 2 κ + 1 | ≤ | η | B 1 M 1 ; 0 < B 1 < 2 L 1 2 ( 1 + κ ) M 1 | η | | η | B 1 M 1 + 1 + κ 2 − L 1 2 B 1 M 1 | η | 2 η 2 B 1 3 | { ( 1 + κ ) M 1 − 2 ν κ ( 1 + κ ) 2 } η B 1 2 − 2 L 1 2 B 2 | + 2 L 1 2 B 1 ; B 1 ≥ 2 L 1 2 ( 1 + κ ) M 1 | η |$
and
$| d 2 κ + 1 − δ d κ + 1 2 | ≤ | η | B 1 M 1 ; | κ − 2 δ + 1 | < J 1 B 1 3 | κ − 2 δ + 1 | | η | 2 | { ( κ + 1 ) M 1 − 2 ν κ ( κ + 1 ) 2 } η B 1 2 − 2 L 1 2 B 2 | ; | κ − 2 δ + 1 | ≥ J 1 ,$
where
$J 1 = { ( 1 + κ ) M 1 − 2 ν κ ( 1 + κ ) 2 } η B 1 2 − 2 L 1 2 B 2 M 1 B 1 2 η ,$
$L 1 = ( 1 + κ ) ( ( κ − 1 ) ν + 1 )$
and
$M 1 = ( 1 + 2 κ ) ( ( 2 κ − 1 ) ν + 1 ) ,$
Remark 3.
If $ν = 0$ and $η = 1$ in Corollary 1 are allowed, then the first and second theorems of Tang et al. [19] are obtained.
Choosing $κ = 1$ in Theorem 1, we have
Corollary 2.
If s $∈ P σ 1 τ ( η , ν , φ )$ is given by (1) and $δ ∈ R$, then
$| d 2 | ≤ | η | B 1 B 1 | { M 2 + 2 ( N 2 τ ( τ − 1 ) + 2 ν ( 1 − 2 τ ) ) } η B 1 2 − L 2 2 B 2 | + L 2 2 B 1 , | d 3 | ≤ | η | B 1 M 2 ; 0 < B 1 < L 2 2 | η | M 2 | η | B 1 M 2 + 1 − L 2 2 | η | B 1 M 2 η 2 B 1 3 | { M 2 + 2 ( N 2 τ ( τ − 1 ) + 2 ν ( 1 − 2 τ ) ) } η B 1 2 − L 2 2 B 2 | + L 2 2 B 1 ; B 1 ≥ L 2 2 | η | M 2 ,$
and
$| d 3 − δ d 2 2 | ≤ | η | B 1 M 2 ; | 1 − δ | < J 2 | η | 2 B 1 3 | 1 − δ | | { M 2 + 2 ( N 2 τ ( τ − 1 ) + 2 ν ( 1 − 2 τ ) ) } η B 1 2 − L 2 2 B 2 | ; | 1 − δ | ≥ J 2 ,$
where
$J 2 = { M 2 + 2 ( N 2 τ ( τ − 1 ) + 2 ν ( 1 − 2 τ ) ) } η B 1 2 − L 2 2 B 2 η M 2 B 1 2 ,$
$L 2 = 2 ( ( 1 + ν ) τ − ν ) ,$
$M 2 = 3 ( ( 1 + 2 ν ) τ − ν )$
and
$N 2 = 3 ν + 1 .$
Setting $η = τ = 1$ in Corollary 2, we obtain the following.
Corollary 3.
If $s ∈$$P σ 1 1 ( 1 , ν , φ )$ is given by (1) and $δ ∈ R$, then
$| d 2 | ≤ B 1 B 1 | ( 3 − ν ) B 1 2 − 4 B 2 | + 4 B 1 , | d 3 | ≤ B 1 3 ( ν + 1 ) ; 0 < B 1 < 4 3 ( ν + 1 ) B 1 3 ( ν + 1 ) + 1 − 4 3 ( ν + 1 ) B 1 B 1 3 | ( 3 − ν ) B 1 2 − 4 B 2 | + 4 B 1 ; B 1 ≥ 4 3 ( ν + 1 ) ,$
and
$| d 3 − δ d 2 2 | ≤ B 1 3 ( ν + 1 ) ; | 1 − δ | < ( 3 − ν ) B 1 2 − 4 B 2 3 ( ν + 1 ) B 1 2 B 1 3 | 1 − δ | | ( 3 − ν ) B 1 2 − 4 B 2 | ; | 1 − δ | ≥ ( 3 − ν ) B 1 2 − 4 B 2 3 ( ν + 1 ) B 1 2 .$
Remark 4.
When $ν = 0$ is selected in Corollary 3, we obtain Corollaries 1 and 4 of Tang et al. [19] (also see [30]).

3. The Class $Q σ κ τ ( η , ν , ϱ )$

Let $φ ( ς ) = 1 + 2 ϱ ς + 2 ϱ 2 ς 2 + ⋯ = 1 + ς 1 − ς ϱ$ in Definition 1. Then, we have the subclass of all $s ∈ σ κ$ satisfying
$a r g 1 η ν { ( ς s ′ ( ς ) ) ′ } τ s ′ ( ς ) + ( 1 − ν ) ( s ′ ( ς ) ) τ − 1 + 1 < ϱ π 2$
and
$a r g 1 η ν { ( ϰ g ′ ( ϰ ) ) ′ } τ g ′ ( ϰ ) + ( 1 − ν ) ( g ′ ( ϰ ) ) τ − 1 + 1 < ϱ π 2 ,$
where $g = s − 1$, $0 < ϱ ≤ 1$, $η ∈ C * , τ ≥ 1$, and $0 ≤ ν < 1$. We denote this class by $Q σ κ τ ( η , ν , ϱ ) = P σ κ τ ( η , ν , 1 + ς 1 − ς ϱ )$.
Remark 5. (i) The family $Q σ κ τ ( η , 0 , ϱ ) ≡ B σ κ τ ( η , ϱ )$, and was explored in [24], where $η ∈ C *$, $τ ≥ 1$ and $0 < ϱ ≤ 1$.
(ii) $Q σ κ 1 ( η , ν , ϱ ) ≡ D σ κ ( η , ν , ϱ )$ is the subfamily of all s $∈ σ κ$ satisfying
$a r g 1 η ν ( ς s ′ ( ς ) ) ′ s ′ ( ς ) + ( 1 − ν ) s ′ ( ς ) − 1 + 1 < ϱ π 2$
and its inverse $g = s − 1$ satisfies
$a r g 1 η ν ( ϰ g ′ ( ϰ ) ) ′ g ′ ( ϰ ) + ( 1 − ν ) g ′ ( ϰ ) − 1 + 1 < ϱ π 2 ,$
where $0 < ϱ ≤ 1$, $η ∈ C *$ and $0 ≤ ν < 1$.
Taking $φ ( ς ) = 1 + ς 1 − ς ϱ$ in Theorem 1, we obtain
Corollary 4.
If the function s given by (3) belongs to the family $∈ Q σ κ τ ( η , ν , ϱ )$ and $δ ∈ R$, then
$| d κ + 1 | ≤ 2 ϱ | η | ϱ | { M ( 1 + κ ) + [ N τ ( τ − 1 ) + ( 1 − ( 1 + κ ) τ ) 2 ν ] ( 1 + κ ) 2 } η − L 2 | + L 2 , | d 2 κ + 1 | ≤ 2 ϱ | η | M ; 0 < ϱ < L 2 M ( 1 + κ ) | η | 2 ϱ | η | M + 1 + κ − L 2 ϱ M | η | 2 ϱ 2 η 2 ϱ | { ( 1 + κ ) M + [ N τ ( τ − 1 ) + ( 1 − ( 1 + κ ) τ ) 2 ν ] ( 1 + κ ) 2 } η − L 2 | + L 2 ; ϱ ≥ L 2 M ( 1 + κ ) | η | ,$
and
$| d 2 κ + 1 − δ d κ + 1 2 | ≤ 2 ϱ | η | M ; | κ − 2 δ + 1 | < J 3 2 ϱ | κ − 2 δ + 1 | | η | 2 | { ( 1 + κ ) M + [ τ ( τ − 1 ) N + ( 1 − ( κ + 1 ) τ ) 2 ν ] ( κ + 1 ) 2 } η − L 2 | ; | κ − 2 δ + 1 | ≥ J 3 ,$
where
$J 3 = { M ( 1 + κ ) + [ N τ ( τ − 1 ) + ( 1 − ( 1 + κ ) τ ) 2 ν ] ( 1 + κ ) 2 } η − L 2 η M ,$
L, M, and N are as in (13), (14), and (15), respectively.
Remark 6.
For $ν = 0$ in Corollary 4, we obtain Corollary 4 in [24].
Choosing $τ = 1$ in $Q σ κ τ ( η , ν , φ )$, we obtain the corollary given below:
Corollary 5.
If the function s given by (3) belongs to the family $D σ κ ( η , ν , ϱ )$ and $δ ∈ R$, then
$| d κ + 1 | ≤ 2 ϱ | η | ϱ | { M 1 ( 1 + κ ) − 2 ν κ ( 1 + κ ) 2 } η − L 1 2 | + L 1 2 , | d 2 κ + 1 | ≤ 2 ϱ | η | M 1 ; 0 < ϱ < L 1 2 M 1 ( 1 + κ ) | η | 2 ϱ | η | M 1 + 1 + κ − L 1 2 ϱ M 1 | η | 2 ϱ 2 η 2 ϱ | { M 1 ( 1 + κ ) − 2 ν κ ( 1 + κ ) 2 } η − L 1 2 | + L 2 ; ϱ ≥ L 1 2 ( 1 + κ ) M 1 | η |$
and
$| d 2 κ + 1 − δ d κ + 1 2 | ≤ 2 ϱ | η | M 1 ; | κ − 2 δ + 1 | < J 4 2 ϱ | κ − 2 δ + 1 | | η | 2 | { M 1 ( κ + 1 ) − 2 ν κ ( κ + 1 ) 2 } η − L 1 2 | ; | κ − 2 δ + 1 | ≥ J 4 ,$
where
$J 4 = { M 1 ( 1 + κ ) − 2 ν κ ( 1 + κ ) 2 } η − L 1 2 η M 1 ,$
$L 1$ and $M 1$ are as in (30) and (31), respectively.
Corollary 4 yields the following if $κ = 1$:
Corollary 6.
If s $∈ Q σ 1 τ ( η , ν , ϱ )$ is given by (1) and $δ ∈ R$, then
$| d 2 | ≤ 2 | η | ϱ ϱ | { 2 M 2 + 4 ( N 2 τ ( τ − 1 ) + 2 ν ( 1 − 2 τ ) ) } η − L 2 2 | + L 2 2 , | d 3 | ≤ 2 | η | ϱ M 2 ; 0 < ϱ < L 2 2 2 | η | M 2 2 | η | ϱ M 2 + 2 − L 2 2 | η | ϱ M 2 2 η 2 ϱ 2 ϱ | { 2 M 2 + 4 ( N 2 τ ( τ − 1 ) + 2 ν ( 1 − 2 τ ) ) } η − L 2 2 | + L 2 2 ; ϱ ≥ L 2 2 2 | η | M 2 ,$
and
$| d 3 − δ d 2 2 | ≤ 2 | η | ϱ M 2 ; | 1 − δ | < J 5 2 | η | 2 ϱ | 1 − δ | | { 2 M 2 + 4 ( N 2 τ ( τ − 1 ) + 2 ν ( 1 − 2 τ ) ) } η − L 2 2 | ; | 1 − δ | ≥ J 5 ,$
where
$J 5 = { 2 M 2 + 4 ( N 2 τ ( τ − 1 ) + 2 ν ( 1 − 2 τ ) ) } η − L 2 2 2 η M 2 ,$
$L 2 , M 2$, and $N 2$ are as in (32), (33), and (34), respectively.
Corollary 6 would yield the following if $η = τ = 1$.
Corollary 7.
If the function s of the form (1) $∈ Q σ 1 1 ( 1 , ν , φ )$ and $δ ∈ R$, then
$| d 2 | ≤ ϱ 2 ϱ ( 1 − ν ) + 2 , | d 3 | ≤ 2 ϱ 3 ( ν + 1 ) ; 0 < ϱ < 2 3 ( ν + 1 ) 2 ϱ 3 ( ν + 1 ) + 1 − 2 3 ϱ ( ν + 1 ) 2 ϱ 2 ϱ ( 1 − ν ) + 2 ; ϱ ≥ 2 3 ( ν + 1 ) ,$
and
$| d 3 − δ d 2 2 | ≤ 2 ϱ 3 ( ν + 1 ) ; | 1 − δ | < 1 − ν 3 ( ν + 1 ) ϱ | 1 − δ | 1 − ν ; | 1 − δ | ≥ 1 − ν 3 ( ν + 1 ) .$
Remark 7.
Letting $ν = 0$ in Corollary 7, we obtain Corollary 2 of Tang et al. [19]. The estimate obtained here for $| d 3 |$ is more accurate when compared to that in Theorem 2 of Srivastava et al. [9].

4. The Class $X σ κ τ ( η , ν , ξ )$

If $φ ( ς ) = 1 + 2 ( 1 − ξ ) ς + 2 ( 1 − ξ ) ς 2 + ⋯ = 1 + ( 1 − 2 ξ ) ς 1 − ς$ in Definition 1, then we have the subset of all $s ∈ σ κ$ satisfying
$R ν { ( ς s ′ ( ς ) ) ′ } τ s ′ ( ς ) + ( 1 − ν ) ( s ′ ( ς ) ) τ − 1 1 η + 1 > ξ$
and
$R ν { ( ϰ g ′ ( ϰ ) ) ′ } τ g ′ ( ϰ ) + ( 1 − ν ) ( g ′ ( ϰ ) ) τ − 1 1 η + 1 > ξ ,$
where $g = s − 1 , η ∈ C * , 0 ≤ ξ < 1 , τ ≥ 1$, and $0 ≤ ν < 1$. We denote this set by $X σ κ τ ( η , ν , ξ ) = P σ κ τ ( η , ν , 1 + ( 1 − 2 ξ ) ς 1 − ς$.
Remark 8. (i). The family $X σ κ τ ( η , 0 , ξ ) ≡ E σ κ τ ( η , ξ )$, $τ ≥ 1 , 0 ≤ ξ < 1$, and was studied in [24].(ii). $X σ κ 1 ( η , ν , ξ ) ≡ F σ κ ( η , ν , ξ )$ is a set of all s $∈ σ κ$ satisfying
$R ν ( ς s ′ ( ς ) ) ′ s ′ ( ς ) + ( 1 − ν ) s ′ ( ς ) − 1 1 η + 1 > ξ$
and its inverse $g = s − 1$ satisfies
$R ν ( ϰ g ′ ( ϰ ) ) ′ g ′ ( ϰ ) + ( 1 − ν ) g ′ ( ϰ ) − 1 1 η + 1 > ξ ,$
where $η ∈ C * , 0 ≤ ξ < 1$, and $0 ≤ ν < 1$.
Allowing $φ ( ς ) = 1 + ( 1 − 2 ξ ) ς 1 − ς$, $0 ≤ ξ < 1 ,$ in Theorem 1, we obtain
Corollary 8.
Let the function s of the form (3) belong to the class $X σ κ τ ( η , ν , ξ )$ and $δ ∈ R$. Then,
$| d κ + 1 | ≤ ( 1 − ξ ) 2 | η | | { ( 1 + κ ) M + ( 1 + κ ) 2 [ N τ ( τ − 1 ) + ( 1 − ( 1 + κ ) τ ) 2 ν ] } ( 1 − ξ ) η − L 2 | + L 2 ,$
$| d 2 κ + 1 | ≤$
$( 1 − ξ ) 2 | η | M ; 1 − L 2 ( 1 + κ ) M | η | < ξ < 1 ( 1 − ξ ) 2 | η | M + 1 + κ − L 2 ( 1 − ξ ) M | η | 2 ( 1 − ξ ) 2 | η | 2 | { ( 1 + κ ) M + [ N τ ( τ − 1 ) + ( 1 − ( 1 + κ ) τ ) 2 ν ] ( 1 + κ ) 2 } ( 1 − ξ ) η − L 2 | + L 2 ; 0 ≤ ξ ≤ 1 − L 2 ( 1 + κ ) M | η |$
and
$| d 2 κ + 1 − δ d κ + 1 2 | ≤ ( 1 − ξ ) 2 | η | M ; | κ − 2 δ + 1 | < J 6 2 ( 1 − ξ ) 2 | η | 2 | κ − 2 δ + 1 | | { ( 1 + κ ) M + [ N τ ( τ − 1 ) + ( 1 − ( 1 + κ ) τ ) 2 ν ] ( 1 + κ ) 2 } ( 1 − ξ ) η − L 2 | ; | κ − 2 δ + 1 | ≥ J 6 ,$
where
$J 6 = { ( 1 + κ ) M + ( 1 + κ ) 2 [ N τ ( τ − 1 ) + ( 1 − ( 1 + κ ) τ ) 2 ν ] } ( 1 − ξ ) η − L 2 M ( 1 − ξ ) η .$
L, M, and N are as in (13), (14), and (15), respectively.
Remark 9.
We obtain Corollary 7 of Aldawish et al. [24] if $ν = 0$ in Corollary 8. In addition, we obtain Corollary 11 of Swamy et al. [22] when $η = τ = 1$.
Corollary 9.
Let the function s of the form (3) belong to the class $F σ κ τ ( η , ν , ξ )$ and $δ ∈ R$. Then,
$| d κ + 1 | ≤ ( 1 − ξ ) 2 | η | | { ( 1 + κ ) M 1 − 2 ν κ ( 1 + κ ) 2 } ( 1 − ξ ) η − L 1 2 | + L 1 2 , | d 2 κ + 1 | ≤ { ( 1 − ξ ) 2 | η | M 1 ; 1 − L 1 2 ( κ + 1 ) M 1 | η | < ξ < 1 ( 1 − ξ ) 2 | η | M 1 + 1 + κ − L 1 2 ( 1 − ξ ) M 1 | η | 2 ( 1 − ξ ) 2 | η | 2 | { ( 1 + κ ) M 1 − 2 ν κ ( 1 + κ ) 2 } η ( 1 − ξ ) − L 1 2 | + L 1 2 ; 0 ≤ ξ ≤ 1 − L 1 2 ( 1 + κ ) M 1 | η |$
and
$| d 2 κ + 1 − δ d κ + 1 2 | ≤ 2 | η | ( 1 − ξ ) M 1 ; | κ − 2 δ + 1 | < J 7 2 | η | 2 ( 1 − ξ ) 2 | κ − 2 δ + 1 | | { ( 1 + κ ) M 1 − 2 ν κ ( 1 + κ ) 2 } ( 1 − ξ ) η − L 1 2 | ; | κ − 2 δ + 1 | ≥ J 7 ,$
where
$J 7 = { ( 1 + κ ) M 1 − 2 ν κ ( 1 + κ ) 2 } ( 1 − ξ ) η − L 1 2 M 1 ( 1 − ξ ) η ,$
$L 1$ and $M 1$ are as in (30) and (31), respectively.
If we let $κ = 1$ in Corollary 8, then we have
Corollary 10.
Let the function s of the form (1) belong to the class $X σ 1 τ ( η , ν , ξ )$ and $δ ∈ R$. Then,
$| d 2 | ≤ 2 ( 1 − ξ ) | η | | { 2 M 2 + 4 ( N 2 τ ( τ − 1 ) + 2 ν ( 1 − 2 τ ) ) } ( 1 − ξ ) η − L 2 2 | + L 2 2 , | d 3 | ≤ 2 ( 1 − ξ ) | η | M 2 ; 1 − L 2 2 2 M 2 | η | < ξ < 1 2 ( 1 − ξ ) | η | M 2 + 2 − L 2 2 ( 1 − ξ ) M 2 | η | 2 ( 1 − ξ ) 2 | η | 2 | { 2 M 2 + 4 ( N τ ( τ − 1 ) + 2 ν ( 1 − 2 τ ) ) } ( 1 − ξ ) η − L 2 2 | + L 2 2 ; 0 ≤ ξ ≤ 1 − L 2 2 2 M 2 | η |$
and
$| d 3 − δ d 2 2 | ≤ 2 ( 1 − ξ ) | η | 3 ( ν + 1 ) ; | 1 − δ | < J 8 2 ( 1 − ξ ) 2 | η | 2 | 1 − δ | | { 2 M 2 + 4 ( N 2 τ ( τ − 1 ) + 2 ν ( 1 − 2 τ ) ) } ( 1 − ξ ) η − L 2 2 | ; | 1 − δ | ≥ J 8 ,$
where
$J 8 = { 2 M 2 + 4 ( N τ ( τ − 1 ) + 2 ν ( 1 − 2 τ ) ) } ( 1 − ξ ) η − L 2 2 2 M 2 ( 1 − ξ ) η ,$
$L 2 , M 2$, and $N 2$ are as in (32), (33), and (34), respectively.
If $η = τ = 1$ in Corollary 10, then we obtain
Corollary 11.
If s $∈ X σ 1 1 ( 1 , ν , φ )$ is of the form (1) and $δ ∈ R$, then
$| d 2 | ≤ 2 ( 1 − ξ ) | ( 3 − ν ) ( 1 − ξ ) − 2 | + 2 , | d 3 | ≤ 2 ( 1 − ξ ) 3 ( 1 + ν ) ; 3 ν + 1 3 ( 1 + ν ) < ξ < 1 2 ( 1 − ξ ) 3 ( 1 + ν ) + 1 − 2 3 ( 1 − ξ ) ( 1 + ν ) 2 ( 1 − ξ ) 2 | ( 1 − ξ ) ( 3 − ν ) − 2 | + 2 ; 0 ≤ ξ ≤ 3 ν + 1 3 ( 1 + ν ) ,$
and
$| d 3 − δ d 2 2 | ≤ 2 ( 1 − ξ ) 3 ( 1 + ν ) ; | 1 − δ | < ( 1 − ξ ) ( 3 − ν ) − 2 3 ( 1 − ξ ) ( 1 + ν ) 2 | 1 − δ | ( 1 − ξ ) 2 | ( 1 − ξ ) ( 3 − ν ) − 2 | ; | 1 − δ | ≥ ( 3 − ν ) ( 1 − ξ ) − 2 3 ( 1 − ξ ) ( 1 + ν ) ,$
Remark 10.
Putting $ν = 0$ in Corollary 11, we obtain Corollary 3 of Tang et al. [19]. The estimates obtained here for $| d 2 |$ and $| d 3 |$ are more accurate when compared to those estimates of Theorem 2 in [9].

5. Conclusions

In this paper, a new class $P σ κ τ ( η , ν , φ )$ is explored and the upper bounds of $| d κ + 1 |$, $| d 2 κ + 1 |$, and $| d 2 κ + 1 − δ d κ + 1 2 |$ $δ ∈ ℜ ,$ are estimated for elements in $P σ κ τ ( η , ν , φ )$. Two special cases $Q σ κ ϱ ( η , ν , τ ) = P σ κ τ ( η , ν , 1 + ς 1 − ς ϱ ) , 0 < ϱ ≤ 1$ and $X σ κ ξ ( η , ν , τ ) = P σ κ τ ( η , ν , 1 + ( 1 − 2 ξ ) ς 1 − ς )$, $0 ≤ ξ < 1$, have been considered. In addition, we have uncovered pertinent links to previous results and given a few observations. This paper could inspire researchers towards further investigations using the (i) integro-differential operator [31], (ii) q-differential operator [32], (iii) q-integral operator [33], and (iv) Hohlov operator [34].

Author Contributions

S.R.S. and L.-I.C.: implementation and original draft; S.R.S.: analysis, methodology, software, and conceptualization; L.-I.C.: validation, resources, and editing. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Not applicable.

Acknowledgments

The authors are appreciative of the reviewers who provided insightful criticism, suggestions, and counsel that helped them to modify and enhance the paper’s final version.

Conflicts of Interest

The authors declare no conflict of interest.

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Swamy, S.R.; Cotîrlă, L.-I. A New Pseudo-Type κ-Fold Symmetric Bi-Univalent Function Class. Axioms 2023, 12, 953. https://doi.org/10.3390/axioms12100953

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Swamy SR, Cotîrlă L-I. A New Pseudo-Type κ-Fold Symmetric Bi-Univalent Function Class. Axioms. 2023; 12(10):953. https://doi.org/10.3390/axioms12100953

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Swamy, Sondekola Rudra, and Luminita-Ioana Cotîrlă. 2023. "A New Pseudo-Type κ-Fold Symmetric Bi-Univalent Function Class" Axioms 12, no. 10: 953. https://doi.org/10.3390/axioms12100953

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