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Article

On a New Class of Concave Bi-Univalent Functions Associated with Bounded Boundary Rotation

by
Prathviraj Sharma
1,
Srikandan Sivasubramanian
1,*,
Gangadharan Murugusundaramoorthy
2 and
Nak Eun Cho
3,*
1
Department of Mathematics, University College of Engineering Tindivanam, Anna University, Tindivanam 604001, Tamilnadu, India
2
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, Tamilnadu, India
3
Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea
*
Authors to whom correspondence should be addressed.
Mathematics 2025, 13(3), 370; https://doi.org/10.3390/math13030370
Submission received: 28 December 2024 / Revised: 20 January 2025 / Accepted: 22 January 2025 / Published: 23 January 2025

Abstract

:
In this research article, we introduce a new subclass of concave bi-univalent functions associated with bounded boundary rotation defined on an open unit disk. For this new class, we make an attempt to find the first two initial coefficient bounds. In addition, we investigate the very famous Fekete–Szegö inequality for functions belonging to this new subclass of concave bi-univalent functions related to bounded boundary rotation. For some particular choices of parameters, we derive the earlier estimates on the coefficient bounds, which are stated at the end.

1. Introduction

Indicate with A the family of holomorphic functions which are defined in the open unit disk
U d = ξ : | ξ | < 1
of the form
g ( ξ ) = ξ + m = 2 b m ξ m .
and normalized by the conditions g ( 0 ) = g ( 0 ) 1 = 0 . Let S A denote the class of univalent functions in the open unit disk U d . For every g S , the Koebe one-quarter theorem confirms that the image of every univalent function g on the open unit disk U d contains a unit ball with its center at the origin and a radius of 1 / 4 . Therefore, every g S has inverse (denote by g 1 ), which satisfies
g 1 ( g ( ξ ) ) = ξ , ξ U d
and
ω = g ( g 1 ( ω ) ) , r | ω | ; 1 4 r .
For each g S , we define the inverse of g :
g 1 ( ω ) = φ ( ω ) = ω + m = 2 φ m ω m ,
where
φ 2 = b 2 , φ 3 = ( 2 b 2 2 b 3 ) φ 4 = ( 5 b 2 3 5 b 2 b 3 + b 4 ) .
If both g and g 1 are univalent in U d , then we say the function g is bi-univalent in U d . The class of all bi-univalent functions is denoted by σ . The functions defined by
log 1 ξ 1 + ξ , log ( 1 ξ ) a n d ξ 1 ξ
are in the class σ with the corresponding inverse functions
1 e 2 ω 1 + e 2 ω , 1 e ω a n d ω 1 + ω .
But the function 2 ξ ξ 2 2 and ξ 1 ξ 2 does not belongs to class σ .
Brannan and Taha [1] studied the class of bi-univalent functions. Furthermore, a few specific subclasses of the bi-univalent function class σ have been established by Brannan and Taha [1]. The first mathematician to explore bi-univalent class functions was Lewin [2], who demonstrated that | b 2 | 1.51 in 1967. Later, Srivastava et al. [3] gave an overview of the class σ with some fascinating examples and presented the first two initial coefficient estimates of | b 2 | and | b 3 | for bi-starlike functions, which were discovered in [3] (also see [4]). These subclasses are comparable to the widely used subclasses bi-starlike, bi-convex, and bi-close-to-convex. Comparably, Li et al. [5], Orhan et al. [6] and Sharma et al. [7] presented specific subclasses of bi-univalent functions linked to bounded boundary rotation. These subclasses are frequently employed as bi-starlike, bi-convex, bi-quasi convex, and bi-close-to-convex with bounded boundary rotation (for additional information, see [6,8,9] and references cited therein). In this study, we present a novel subclass of σ related to bounded boundary rotation, which we refer to as a concave bi-univalent function.

1.1. Concave Function

A domain D is said to be a concave domain if the closed set C D is convex and unbounded. Hence, D is simply connected. A function g : D C { } is said to be concave, if g ( D ) is a concave domain. The analytic description of the above class is given by
C 0 = g S : 1 + ξ g ( ξ ) g ( ξ ) < 0 , ξ U d .
In 2006, Cruz and Pommerenke [10] and many other authors (see [11,12,13,14]) studied concave univalent functions.
A function g : U d C is said to be a concave univalent function with an opening angle ϑ π , 1 < ϑ 2 at infinity if g satisfies the given conditions:
  • g A and satisfies the condition g ( 1 ) = .
  • g maps U d conformally onto a concave set, i.e., g maps U d conformally, a set whose complement with C is convex.
  • The opening angle of the image of f (i.e., g ( U d ) ) at ∞ is equal to or less than ϑ π .
Let us denote the class of all concave univalent functions with an opening angle of ϑ π , as C 0 ϑ . In 2009, Bhowmik [15] pointed out that an analytic function g is concave univalent with an opening angle of ϑ π , if and only if
2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) > 0 , ξ U d .
Sakar and Güney [16] extended the class C 0 ϑ by introducing an order ρ , 0 ρ < 1 , called the class of all concave univalent functions of order ρ , 0 ρ < 1 with an opening angle of ϑ π , as follows:
For α [ 1 , 2 ) , 0 ρ < 1 , f C 0 ϑ ( ρ ) , if and only if P g ( ξ ) > ρ , ξ U d where
P g ( ξ ) = 2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) , ξ U d , g ( 0 ) = g ( 0 ) 1 = 0 .

1.2. Function with Bounded Variation

In 1975, Padmanabhan and Parvatham [17] introduced the class P κ ( ρ ) . For 2 κ and 1 > ρ 0 , let us indicate P κ ( ρ ) as the class of functions q, which are holomorphic and normalized by q ( 0 ) = 1 and q ( 0 ) > 1 , satisfy the condition
0 2 π ( q ( ξ ) ) ρ 1 ρ d t κ π ,
where ξ = r e i t U d . For ρ = 0 , the class P κ ( ρ ) converts into the class P κ , which is defined by Pinchuk [18]. The class P κ denotes the class of functions q ( 0 ) = 1 that are analytic in U d and with a representation of
q ( ξ ) = 1 2 0 2 π e i t + ξ e i t ξ d Λ ( t ) ,
where Λ is a function of bounded variation (which is real-valued) and satisfies
0 2 π d Λ ( t ) = 2 and 0 2 π d Λ ( t ) κ , 2 κ .
If we choose ρ = 0 and κ = 2 , then the class P κ ( ρ ) reduces to the class P , which is known as the class of Carathéodory functions. An interesting observation between the class P κ and Carathéodory functions is that the function q ( ξ ) P κ if there exist two Carathéodory functions, q 1 ( ξ ) and q 2 ( ξ ) , such that
p ( z ) = κ 4 + 1 2 q 1 ( ξ ) κ 4 1 2 q 2 ( ξ ) .
Let V k represent the class of analytic functions g in U d with g ( 0 ) = 0 , g ( 0 ) = 1 , satisfying
1 + ξ g ( ξ ) g ( ξ ) P k .
In 1931, Paatero [19] proved that a function g V κ given in the form (1) maps U d conformally onto an image domain f ( U d ) at most κ π . Paatero [19] has shown that g V κ if and only if
g ( ξ ) = exp 0 2 π log e i t e i t ξ d Λ ( t ) ,
where Λ is a function of bounded variation (which is real-valued) and satisfies
0 2 π d Λ ( t ) = 2 and 0 2 π d Λ ( t ) κ , κ 2 .
Brannan [20] showed that the function g V κ if there exist two starlike functions, f 1 ( ξ ) and f 2 ( ξ ) , such that
f ( ξ ) = f 2 ( ξ ) ξ κ 4 + 1 2 f 1 ( ξ ) ξ κ 4 1 2 .
Paatero [19] gave the following distortion bounds: for the function, g V κ for | ξ | = r < 1
( 1 r ) κ 2 1 ( 1 + r ) κ 2 + 1 | g ( ξ ) | ( 1 + r ) κ 2 + 1 ( 1 r ) κ 2 1 .
In 1971, Pinchuk [18] introduced and studied the class R κ , that is, g R κ if and only if
g ( ξ ) = ξ exp 0 2 π log e i t e i t ξ d Λ ( t ) ,
where Λ is a function of bounded variation (which is real-valued) and satisfies
0 2 π d Λ ( t ) = 2 and 0 2 π d Λ ( t ) κ , κ 2 .
Thus, R κ is the class of functions of bounded radius rotation bounded by κ π .
Lemma 1
([21]). Let an analytic function q ( ξ ) = 1 + m = 1 q m ξ m , ξ U d belong to the class P κ ( ρ ) . Then,
q m κ ( 1 ρ ) , m 1 .
In this study, we present a new subclass of concave bi-univalent functions related to bounded boundary rotation as given in Definition 1. We obtain the first two initial bounds, | a 2 | and | a 3 | , and also find the well-known Fekete–Szegö inequality | a 3 a 2 2 | for functions in this new class. For specific parameter selections, the previous findings are pointed out as corollaries, which have not been discussed so far.

2. Examples for the Class Concave Functions with Bounded Boundary Rotation

2.1. Concave Functions with Bounded Boundary Rotation

Suppose 1 < ϑ 2 and 4 κ 2 . A function g given in the form (1) is said to be a concave function with bounded boundary rotation if the function g satisfies the following condition:
2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) P κ .
The class of all concave functions with bounded boundary rotation is denoted by CO ϑ ( κ ) .

2.2. Integral Representation of CO ϑ ( κ )

Theorem 1.
Suppose 1 < ϑ 2 and 4 κ 2 . If a function g CO ϑ ( κ ) , then
g ( ξ ) = ( 1 ξ ) ( ϑ + 1 ) exp 1 2 0 2 π ( ϑ 1 ) log ( 1 ξ e i t ) d Λ ( t ) ,
where Λ is a function of bounded variation (which is real-valued) and satisfies
0 2 π d Λ ( t ) = 2 a n d 0 2 π d Λ ( t ) κ , 2 κ .
Proof. 
Since g CO ϑ ( κ ) , there exists an analytic function r ( ξ ) belonging to the class P κ such that
2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) = r ( ξ ) .
Hence, from (5), we obtain
ϑ + 1 1 ξ g ( ξ ) g ( ξ ) = ϑ 1 2 r ( ξ ) 1 ξ .
Since r ( ξ ) P κ ,
r ( ξ ) 1 ξ = 1 ξ 1 2 0 2 π 1 + e i t ξ 1 e i t ξ d Λ ( t ) 1 , = 1 2 ξ 0 2 π 1 + e i t ξ 1 e i t ξ d Λ ( t ) 0 2 π d Λ ( t ) , = 0 2 π e i t 1 e i t ξ d Λ ( t ) .
Therefore,
0 ξ r ( ζ ) 1 ζ d ζ = 0 2 π log ( 1 e i t ξ ) d Λ ( t ) .
From (6) and (7), we obtain (4). This completes the proof of Theorem 1. □

2.3. Relation Between Class CO ϑ ( κ ) and V κ

Theorem 2.
Suppose 1 < ϑ 2 and 4 κ 2 . If the function g CO ϑ ( κ ) , then there exists a function ψ V κ such that
g ( ξ ) = [ ψ ( ξ ) ] ϑ + 1 2 ( 1 ξ ) ϑ + 1 .
Proof. 
Since g CO ϑ ( κ ) , there exists an analytic function r ( ξ ) belonging to the class P κ such that
2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) = r ( ξ ) .
Since r P κ , there exists a function ψ V κ such that
r ( ξ ) = 1 + ξ ψ ( ξ ) ψ ( ξ ) .
Hence, from (9) and (10), we obtain
2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) = 1 + ξ ψ ( ξ ) ψ ( ξ ) .
Equation (11) can be written as
ϑ + 1 1 ξ g ( ξ ) g ( ξ ) = ϑ + 1 2 ψ ( ξ ) ψ ( ξ ) .
Integrating (12), we obtain (8). This completes the proof of Theorem 2. □

2.4. Examples

Let us define some examples of functions belonging to the class CO ϑ ( κ ) .
Example 1.
The function g 1 : U d C defined by
g 1 ( ξ ) = 0 ξ e 1 ϑ 2 t ( t 1 ) ϑ + 1 d t
is in the class CO ϑ ( κ ) . Since
g 1 ( ξ ) = e 1 ϑ 2 ξ ( ξ 1 ) ϑ + 1
we have
2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) = 1 + ( ξ ) .
Therefore,
0 2 π 2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) = 0 2 π ( 1 + cos θ ) d θ = 2 π κ π .
Hence, g 1 CO ϑ ( κ ) .
Example 2.
The function g 2 : U d C defined by
g 2 ( ξ ) = 0 ξ e κ ( 1 ϑ ) 4 t ( t 1 ) ϑ + 1 d t
is in the class CO ϑ ( κ ) . Since
g 2 ( ξ ) = e κ ( 1 ϑ ) 4 ξ ( ξ 1 ) ϑ + 1 ,
we have
2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) = 1 + κ 2 ( ξ ) .
Therefore,
0 2 π 2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) = 0 2 π ( 1 + κ 2 cos θ ) d θ = 2 π κ π .
Hence, g 2 CO ϑ ( κ ) .
Example 3.
The function g 3 : U d C defined by
g 3 ( ξ ) = 0 ξ e 1 ϑ 4 t ( t 1 ) ϑ + 1 d t
and is in the class CO ϑ ( κ ) . Since
g 3 ( ξ ) = e 1 ϑ 4 ξ ( ξ 1 ) ϑ + 1 ,
we have
2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) = 1 + 1 2 ( ξ ) .
Therefore,
0 2 π 2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) = 0 2 π ( 1 + 1 2 cos θ ) d θ = 2 π κ π .
Hence, g 3 CO ϑ ( κ ) .
Example 4.
The function g 4 : U d C defined by
g 4 ( ξ ) = 0 ξ e κ ( 1 ϑ ) 8 t ( t 1 ) ϑ + 1 d t
belongs to the class CO ϑ ( κ ) . Since
g 4 ( ξ ) = e κ ( 1 ϑ ) 8 ξ ( ξ 1 ) ϑ + 1 ,
we have
2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) = 1 + κ 4 ( ξ ) .
Therefore,
0 2 π 2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) = 0 2 π ( 1 + κ 4 cos θ ) d θ = 2 π κ π .
Hence, g 4 CO ϑ ( κ ) .

3. Concave Bi-Univalent Functions with Bounded Boundary Rotation

Definition 1.
Suppose 1 < ϑ 2 ,   ρ [ 0 , 1 ) and 4 κ 2 . A function g σ given in the form (1) is said to be a bi-concave function with bounded boundary rotation of order ρ if the functions g and g 1 = φ satisfy the following conditions:
2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) P κ ( ρ )
and
2 ϑ 1 ϑ + 1 2 1 ω 1 + ω 1 ω φ ( ω ) φ ( ω ) P κ ( ρ ) .
The class of all concave bi-univalent functions with bounded boundary rotation of order ρ is denoted by CO σ ϑ ( κ , ρ ) .
Remark 1.
(1) Upon fixing ρ = 0 , we have a new class, CO σ ϑ ( κ , ρ ) CO σ ϑ ( κ ) , the class of all bi-concave functions with bounded boundary rotation.
(2) Assuming κ = 2 , the class CO σ ϑ ( κ , ρ ) CO σ ϑ ( ρ ) , which was called the class of all bi-concave functions of order ρ, which was defined by Sakar and Güney [16].
Theorem 3.
Suppose 1 < ϑ 2 ,   ρ [ 0 , 1 ) and 4 κ 2 . If the function g CO σ ϑ ( κ , ρ ) , then
| b 2 | 1 + ϑ 2 + κ ( 1 ρ ) ( ϑ 1 ) 4 ,
| b 3 | 1 + ϑ 2 + κ ( 1 ρ ) ( ϑ 1 ) 4
and for any R that is a real number,
| b 3 b 2 2 | 1 + ϑ 2 + κ ( 1 ρ ) ( ϑ 1 ) 4 ( 1 ) f o r < 2 3 , 1 + ϑ 2 ( 1 ) + κ ( 1 ρ ) ( ϑ 1 ) 12 f o r 2 3 < 1 , 1 + ϑ 2 ( 1 ) + κ ( 1 ρ ) ( ϑ 1 ) 12 f o r 1 < 4 3 , 1 + ϑ 2 + κ ( 1 ρ ) ( ϑ 1 ) 4 ( 1 ) f o r 4 3 .
Proof. 
Since g CO σ ϑ ( κ , ρ ) , from Definition 1, there exist two analytic functions r ( ξ ) , and u ( ω ) belongs to the class P κ ( ρ ) such that
2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) = r ( ξ )
and
2 ϑ 1 ϑ + 1 2 1 ω 1 + ω 1 ω φ ( ω ) φ ( ω ) = u ( ω ) ,
where
r ( ξ ) = 1 + m = 1 r m ξ m = 1 + r 1 ξ + r 2 ξ 2 + .
and
u ( ω ) = 1 + m = 1 u m ω m = 1 + u 1 ω + u 2 ω 2 + .
Since
2 ϑ 1 ϑ + 1 2 1 + ξ 1 ξ 1 ξ g ( ξ ) g ( ξ ) = 1 + 2 ( ϑ + 1 ) ϑ 1 4 b 2 ϑ 1 ξ + 2 ( ϑ + 1 ) ϑ 1 2 ( 6 b 3 4 b 2 2 ) ϑ 1 ξ 2 +
and
2 ϑ 1 ϑ + 1 2 1 ω 1 + ω 1 ω φ ( ω ) φ ( ω ) = 1 2 ( ϑ + 1 ) ϑ 1 4 b 2 ϑ 1 ω + 2 ( ϑ + 1 ) ϑ 1 2 ( 8 b 2 2 6 b 3 ) ϑ 1 ω 2 + ,
from (18)–(21), we obtain
2 ( ϑ + 1 ) ϑ 1 4 b 2 ϑ 1 = r 1 ,
2 ( ϑ + 1 ) ϑ 1 2 ( 6 b 3 4 b 2 2 ) ϑ 1 = r 2 ,
2 ( ϑ + 1 ) ϑ 1 + 4 b 2 ϑ 1 = u 1
and
2 ( ϑ + 1 ) ϑ 1 2 ( 8 b 2 2 6 b 3 ) ϑ 1 = u 2 .
Hence, from (22) and (24), we obtain
4 b 2 ϑ 1 = 2 ( ϑ + 1 ) ϑ 1 r 1 = u 1 + 2 ( ϑ + 1 ) ϑ 1 .
By using Lemma 1 in (26), we obtain
| b 2 | ϑ + 1 2 + κ ( 1 ρ ) ( ϑ 1 ) 4 .
From (23) and (25), we obtain
8 b 2 2 ϑ 1 = 4 ( ϑ + 1 ) ϑ 1 r 2 u 2 .
Hence, by using Lemma 1 in (28), we obtain
| b 2 2 | ϑ + 1 2 + κ ( 1 ρ ) ( ϑ 1 ) 4 .
Equations (27) and (29) give the bound of | b 2 | given in (13). Now, subtracting (22) from (24), we obtain
24 b 3 ϑ 1 = 24 b 2 2 ϑ 1 u 2 + r 2 .
By using (28) in (30), we obtain
12 b 3 ϑ 1 = 6 ( ϑ + 1 ) ϑ 1 2 r 2 u 2 .
Hence, by using Lemma 1 in (31), we obtain
| b 3 | ϑ + 1 2 + κ ( 1 ρ ) ( ϑ 1 ) 4 .
Equation (32) gives the bound of | b 3 | given in (14). For any R and from (28) and (31), we obtain
b 3 b 2 2 ϑ 1 = ( ϑ + 1 ) ( 1 ) 2 ( ϑ 1 ) r 2 ( 4 3 ) 24 u 2 ( 2 3 ) 24 .
Hence, by using Lemma 1 in (33), we obtain
| b 3 b 2 2 | ϑ 1 ( ϑ + 1 ) | 1 | 2 ( ϑ 1 ) + κ ( 1 ρ ) 24 [ | 4 3 | + | 2 3 | ] .
For the different choice of , Equation (34) gives (15). This completes the proof of Theorem 3. □
By fixing ρ = 0 , we state the following result for g CO σ ϑ ( κ ) .
Corollary 1.
Suppose 1 < ϑ 2 and 4 κ 2 . If a function g CO σ ϑ ( κ ) , then
| b 2 | ϑ + 1 2 + κ ( ϑ 1 ) 4 ,
| b 3 | ϑ + 1 2 + κ ( ϑ 1 ) 4
and for any R that is a real number,
| b 3 b 2 2 | ϑ + 1 2 ( ϑ 1 ) + κ 4 ( 1 ) f o r < 2 3 , ϑ + 1 2 ( ϑ 1 ) ( 1 ) + κ 12 f o r 2 3 < 1 , ϑ + 1 2 ( ϑ 1 ) ( 1 ) + κ 12 f o r 1 < 4 3 , ϑ + 1 2 ( ϑ 1 ) + κ 4 ( 1 ) f o r 4 3 .
By taking κ = 2 we obtain the following result for g CO σ ϑ ( ρ ) .
Corollary 2.
Suppose 1 < ϑ 2 and ρ [ 0 , 1 ) . If the function g CO σ ϑ ( ρ ) , then
| b 2 | ( ϑ + 1 ) + ( 1 ρ ) ( ϑ 1 ) 2 ,
| b 3 | ( ϑ + 1 ) + ( 1 ρ ) ( ϑ 1 ) 2
and for any R that is a real number,
| b 3 b 2 2 | ( ϑ + 1 ) + ( 1 ρ ) ( ϑ 1 ) 2 ( 1 ) f o r < 2 3 , 3 ( ϑ + 1 ) ( 1 ) + ( 1 ρ ) ( ϑ 1 ) 6 f o r 2 3 < 1 , 3 ( ϑ + 1 ) ( 1 ) + ( 1 ρ ) ( ϑ 1 ) 6 f o r 1 < 4 3 , ( ϑ + 1 ) + ( 1 ρ ) ( ϑ 1 ) 2 ( 1 ) f o r 4 3 .
For the particular choice of κ = 2 and ρ = 0 , the class CO σ ϑ ( κ , ρ ) converts into the class CO σ ϑ . Hence, we obtain the following result for the functions belonging to the class CO σ ϑ .
Corollary 3.
Suppose 1 < ϑ 2 . If the function g CO σ ϑ , then
| b 2 | ϑ , | b 3 | ϑ
and for any R that is a real number,
| b 3 b 2 2 | ϑ ( 1 ) f o r < 2 3 , 3 ( ϑ + 1 ) ( 1 ) + ( ϑ 1 ) 6 f o r 2 3 < 1 , 3 ( ϑ + 1 ) ( 1 ) + ( ϑ 1 ) 6 f o r 1 < 4 3 , ϑ ( 1 ) f o r 4 3 .

4. Concluding Remarks and Observations

In this research article, we introduce and examine some properties of a new class of concave bi-univalent functions associated with bounded boundary rotation in open unit disk U d . We investigates the initial Taylor–Maclaurin coefficients, | b 2 | and | b 3 | , for functions belonging to concave bi-univalent functions associated with bounded boundary rotation. Note that by using different values of parameters, we obtain certain corollaries, which verifies earlier results on the bounds | b 2 | and | b 3 | which were obtained by Sakar and Güney [16]. In addition, for the first time, we investigate the very famous Fekete–Szegö inequality | b 3 b 2 2 | for g belonging to this new subclass of concave bi-univalent functions related to bounded boundary rotation.

Author Contributions

Conceptualization, S.S., G.M. and N.E.C.; Formal analysis, P.S., S.S. and N.E.C.; Investigation, P.S. and G.M. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

No data were used in this research.

Acknowledgments

The authors would like to thank the referees as well as the Academic Editor for their valuable comments and suggestions on the original manuscript.

Conflicts of Interest

The authors declare no conflicts of interest.

References

  1. Brannan, D.A.; Taha, T.S. On some classes of bi-univalent functions. Studia Univ. Babeş-Bolyai Math. 1986, 31, 70–77. [Google Scholar]
  2. Lewin, M. On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 1967, 18, 63–68. [Google Scholar] [CrossRef]
  3. Srivastava, H.M.; Mishra, A.K.; Gochhayat, P. Certain subclasses of analytic and bi-univalent functions. Appl. Math. Lett. 2010, 23, 1188–1192. [Google Scholar] [CrossRef]
  4. Bulut, S. Coefficient estimates for a class of analytic and bi-univalent functions. Novi Sad J. Math. 2013, 43, 59–65. [Google Scholar]
  5. Li, Y.; Vijaya, K.; Murugusundaramoorthy, G.; Tang, H. On new subclasses of bi-starlike functions with bounded boundary rotation. AIMS Math. 2020, 5, 3346–3356. [Google Scholar] [CrossRef]
  6. Orhan, H.; Magesh, N.A.; Balaji, V.K. Certain classes of bi-univalent functions with bounded boundary variation. Tbilisi Math. J. 2017, 10, 17–27. [Google Scholar] [CrossRef]
  7. Sharma, P.; Sivasubramanian, S.; Cho, N.E. Initial coefficient bounds for certain new subclasses of bi-univalent functions with bounded boundary rotation. AIMS Math. 2023, 8, 29535–29554. [Google Scholar] [CrossRef]
  8. Goswami, P.; Bulut, S.; Sekhawat, N. Coefficients estimates of a new class of analytic bi-univalent functions with bounded boundary rotation. Hacet. J. Math. Stat. 2022, 51, 1271–1279. [Google Scholar] [CrossRef]
  9. Sharma, P.; Sivasubramanian, S.; Cho, N.E. Initial coefficient bounds for certain new subclasses of bi-Bazilevič functions and exponentially bi-convex functions with bounded boundary rotation. Axioms 2024, 13, 25. [Google Scholar] [CrossRef]
  10. Cruz, L.; Pommerenke, C. On concave univalent functions. Complex Var. Elliptic Equ. 2007, 52, 153–159. [Google Scholar] [CrossRef]
  11. Altınkaya, S.; Yalçın, S. General properties of multivalent concave functions involving linear operator of Carlson-Shaffer type. Comptes Rendus Acad. Bulgare Sci. 2016, 69, 1533–1540. [Google Scholar]
  12. Avkhadiev, F.G.; Pommerenke, C.; Wirths, K.-J. Sharp inequalities for the coefficient of concave schlicht functions. Comment. Math. Helv. 2006, 81, 801–807. [Google Scholar] [CrossRef]
  13. Avkhadiev, F.G.; Wirths, K.-J. Concave schlicht functions with bounded opening angle at infinity. Lobachevskii J. Math. 2005, 17, 3–10. [Google Scholar]
  14. Pfaltzgraff, J.A.; Pinchuk, B. A variational method for classes of meromorphic functions. J. Anal. Math. 1971, 24, 101–150. [Google Scholar] [CrossRef]
  15. Bhowmik, B.; Ponnusamy, S.; Wirths, K.-J. Characterization and the pre-Schwarzian norm estimate for concave univalent functions. Monatsh. Math. 2010, 161, 59–75. [Google Scholar] [CrossRef]
  16. Sakar, F.M.; Güney, H. Coefficient estimates for bi-concave functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019, 68, 53–60. [Google Scholar] [CrossRef]
  17. Padmanabhan, K.S.; Parvatham, R. Properties of a class of functions with bounded boundary rotation. Ann. Polon. Math. 1976, 3, 311–323. [Google Scholar] [CrossRef]
  18. Pinchuk, B. Functions of bounded boundary rotation. Isr. J. Math. 1971, 10, 6–16. [Google Scholar] [CrossRef]
  19. Paatero, V. Über Gebiete von beschrankter Randdrehung. Ann. Acad. Sci. Fenn. Ser. 1933, 37, 9. [Google Scholar]
  20. Brannan, D.A. On functions of bounded boundary rotation. I. Proc. Edinb. Math. Soc. 1969, 16, 339–347. [Google Scholar] [CrossRef]
  21. Alkahtani, B.S.T.; Goswami, P.; Bulboacă, T. Estimate for initial MacLaurin coefficients of certain subclasses of bi-univalent functions. Miskolc Math. Notes 2016, 17, 739–748. [Google Scholar] [CrossRef]
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MDPI and ACS Style

Sharma, P.; Sivasubramanian, S.; Murugusundaramoorthy, G.; Cho, N.E. On a New Class of Concave Bi-Univalent Functions Associated with Bounded Boundary Rotation. Mathematics 2025, 13, 370. https://doi.org/10.3390/math13030370

AMA Style

Sharma P, Sivasubramanian S, Murugusundaramoorthy G, Cho NE. On a New Class of Concave Bi-Univalent Functions Associated with Bounded Boundary Rotation. Mathematics. 2025; 13(3):370. https://doi.org/10.3390/math13030370

Chicago/Turabian Style

Sharma, Prathviraj, Srikandan Sivasubramanian, Gangadharan Murugusundaramoorthy, and Nak Eun Cho. 2025. "On a New Class of Concave Bi-Univalent Functions Associated with Bounded Boundary Rotation" Mathematics 13, no. 3: 370. https://doi.org/10.3390/math13030370

APA Style

Sharma, P., Sivasubramanian, S., Murugusundaramoorthy, G., & Cho, N. E. (2025). On a New Class of Concave Bi-Univalent Functions Associated with Bounded Boundary Rotation. Mathematics, 13(3), 370. https://doi.org/10.3390/math13030370

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