Title


Science and Technology Indonesia
e-ISSN:2580-4391 p-ISSN:2580-4405
Vol. 8, No. 2, April 2023

Research Paper

Bi-Univalent Function Classes Defined by Using an Einstein Function and a New
Generalised Operator
Munirah Rossdy1,2*, Rashidah Omar2, Shaharuddin Cik Soh1
1School of Mathematical Sciences, College of Computing, Informatics and Media, Universiti Teknologi MARA, Shah Alam, Selangor, 40450, Malaysia2Mathematical Sciences Studies, College of Computing, Informatics and Media, Universiti Teknologi MARA Sabah Branch, Kota Kinabalu Campus, Kota Kinabalu, Sabah, 88997,
Malaysia
*Corresponding author: munirahrossdy@uitm.edu.my

AbstractLet A be the class of all analytic and univalent functions f (z) = z+Σ∞
k=2

akz
k in the open unit disc 𝔻 = {z:|z|<1 }. S then represents the

classes of every function in A that is univalent in 𝔻. For every f ∈ S, there is an inverse f −1. A function f ∈ A in 𝔻 is categorised asbi-univalent if f and its inverse g = f −1 are both univalent. Motivated by the generalised operator, subordination principle, and thefirst Einstein function, we present a new family of bi-univalent analytic functions on the open unit disc of the complex plane. Thefunctions contained in the subclasses are used to account for the initial coefficient estimate of |a2|. In this study, we derive the resultsfor the covering theorem, distortion theorem, rotation theorem, growth theorem, and the convexity radius for functions of the class
N s,m,k

_ ,𝛼 (Σ, E) of bi-univalent functions related to an Einstein function and a generalised differential operator Ds,m,k_ ,𝛼 f (z). We usethe elementary transformations that preserve the class N s,m,k
_ ,𝛼 (Σ, E) in order to attain the intended results. The required propertiesare then obtained.

KeywordsEinstein Function, Subordination, Distortion Theorem, Covering Theorem, Radius of Convexity, Bi-Univalent Functions
Received: 8 November 2022, Accepted: 13 February 2023
https://doi.org/10.26554/sti.2023.8.2.195-204

1. INTRODUCTION

For the open unit disc 𝔻 = {z ∈ ℂ:|z|<1 }, we set A as denoting
the class of

f (z) = z +
∞∑︁
n=2

anz
n , for z ∈ 𝔻 (1)

analytic functions. S represents the class of all functions
in A that are univalent in 𝔻 (for further detail on univalent
functions, see Duren (2001) ) and satisfy the standard normali-
sation condition f (0) = f ′(0) - 1 = 0. The Koebe one-quarter
theorem (Duren, 2001) demonstrates that each univalent func-
tion f ∈ S having a disc with a radius of 14 possesses an inverse
function f −1 that can be defined by

f −1( f (z)) = z, for z ∈ 𝔻,

and

f ( f −1(w)) = w, |w| < r0( f ), r0( f ) ≤
1
4

The function f ∈ S is deemed as bi-univalent if f and f −1
are both univalent in 𝔻. Let the class of bi-univalent functions
in 𝔻 of the form (1) be denoted by

∑
. Moreover, it is easily

demonstrated that the series expansion of the inverse function
can be written as follows:

g(w) = f −1(w)
= w − a2w2 + (2a22 − a3)w

3 (2)

− (5a32 − 5a2a3 + a4)w
4 + · · · , for w ∈ 𝔻.

Class
∑

includes functions such as z1−z′ − log(1 − z), and
1
2 log(

1+z
1−z) . Nevertheless, the well-known Koebe function does

not belong to
∑

. Other typical instances of functions in S such
as z − z

2

2 , and
z

1−z2 , do not belong to
∑

. For references to
related works on bi-univalent functions, see the revival paper

https://crossmark.crossref.org/dialog/?doi=10.26554/sti.2023.8.2.195-204&amp;domain=pdf
https://doi.org/10.26554/sti.2023.8.2.195-204


Rossdy et. al. Science and Technology Indonesia, 8 (2023) 195-204

by Srivastava et al. (2010) , as well as several other studies (Ali
et al., 2012; Oros and Cotîrlă, 2022; Srivastava et al., 2018).

Normalised analytic function operators are commonly utili-
sed in the field of Geometric Function Theory (GFT), particu-
larly differential and integral operators. A wide range of authors
have written numerous articles on a variety of topics, including
operators and novel generalisation. The differential operator,
which was first introduced in 1975 by Ruscheweyh (1975) , was
a particularly major breakthrough. Differential and integral op-
erators were then presented in a different version by Salagean
(1983) . From there on, many academics have developed new
operators and used them in numerous research topics involving
GFT. They include Rossdy et al. (2022) , Wanas (2019) , Yunus
et al. (2017) , Elhaddad and Darus (2021) , and Frasin (2020) .

In this paper, we provide some information regarding the
differential operator that is applied to examine our new sub-
classes. According to Rossdy et al. (2022) , the differential
operator is defined by:

Definition 1.1 For f ∈ A, 0 < _ < 1, 0 < 𝛼 < 1, m ∈ ℕ =
{1, 2, · · · }, b ∈ ℂ\Z−0 , s ∈ ℂ, k ∈ ℕ0 ,

Ds,m,k
_ ,𝛼 f (z) =z +

∞∑︁
n=2

(
1 + b
n + b

)s
(3)

[1 + _ (n − 1)(1 − _)m]kanzn

We can see that when two functions of the class
∑

are linked
in a convex combination, it need not be bi-univalent. Even
though the two functions of f1(z) = z1−z and f2(z) =

z
1+iz are

examples of bi-univalent functions, their sum, f1 + f2, is not
univalent because its derivative no longer exists at 12 (1 + i).
Nevertheless, several elementary transformations preserve the
class

∑
, as seen below (Wei, 2017; Sivasubramanian et al.,

2014):
i. Rotation: If f ∈

∑
, 0 ∈ ℝ, and g(z) = e−i\ f (ei\z), then

g ∈
∑

;

ii. Dilation: If f ∈
∑

, 0 < r < 1, and g(z)= 1r f (rz), then
g ∈

∑
;

iii. Conjugation: If f ∈
∑

and g(z) = f (z), then g ∈
∑

;

iv. Disk automorphism: If f ∈
∑

, Z ∈ 𝔻, and g(z) =
f ( z+Z

1+Zz
)− f (Z )

(1−|Z |2 ) f ′ (Z ) , then g ∈
∑

.

v. Omitted value transformation: If f ∈
∑

with f (z) ± w
for all z ∈ 𝔻, and g(z)= wf (z)w− f (z) , then g ∈

∑
.

In GFT, determining coefficient estimates |an |(n ∈ ℕ) is es-
sential because this allows details of these functions’ geometric
properties to be obtained. The evaluation of analytic function
coefficients determines the structural characteristics and partic-
ulars of GFT. For example, in the univalent function set, the

second coefficient |a2| implies the covering theorems, growth
and distortion bounds. The renowned Bieberbach Conjecture,
as proven by Louis de Branges (De Branges, 1985) , posits that
the coefficient inequality as written below is true for each f ∈ S
provided by the Taylor-Maclaurin series expansion (1):

|an | ≤ n (n ∈ ℕ\{1}), (4)

where ℕ represents the set of all positive integers. Lewin
in his research Lewin (1967) on bi-univalent functions of the
class

∑
, discovered the bound |a2| < 1.51. Brannan and Clunie

(1980) in their subsequent work proposed that |a2| ≤
√

2.
Additionally, Netanyahu (1969) demonstrated that maxf ∈∑
|a2| =

4
3 . In addition to estimating the coefficients for |a2|

and |a3|, Brannan and Taha (1988) proposed the concepts of
strongly bi-starlike functions of the order 𝛼 and strongly bi-
convex functions of the order 𝛼. Following the lead of Brannan
and Taha (1988) , other researchers (Rossdy et al., 2021; Soni
et al., 2018; Xu et al., 2012) have studied numerous subclasses
of

∑
and determined the coefficient bounds for |a2| and |a3|.

The geometric theory of bi-univalent functions shows more
applications of Lewin’s inequality (Lewin, 1967) . An important
implication is the distortion theorem. This theorem gives non-
sharp upper and lower bounds for |f ′(z)| as f ranges over the
class

∑
.

The idea of subordination is then employed as defined
below:

Definition 1.2 (Miller and Mocanu, 2000) Given that f (z) ≺
g(z), with f being a subordinate to g, and both functions taken
to be analytic. This indicates that f (z) = g(w(z)), where w is
taken as analytic in 𝔻, which corresponds to |w(z)| < 1 and
w(0) = 0.

Ma (1992) presented the subset of functions S∗(𝜙) = {f ∈
A : zf

′ (z)
f (z) ≺ 𝜙(z), 𝜙 ∈ P, z ∈ 𝔻} in 1994, in which the sym-

bol "≺" corresponds to the subordination stated in Definition
1.2 above. Ma (1992) studied several relevant topics, such as
covering, growth, and distortion theorems. Then, by inserting
certain functions for 𝜙 in S∗(𝜙), we obtain various subclasses
of A with distinct geometric analyses, such as those from the
work by Janowski (1970) , Mendiratta et al. (2015) , and Cho
et al. (2019) . Many characteristics of the analytic univalent
functions are connected with differential and integral opera-
tors, such as coefficient bound, covering theorems, distortion
theorems, growth theorems, inclusion properties, and radius
of convexity; all of these have been investigated (Omar and
Abdul Halim, 2012; Zhang et al., 2021; Saheb and Al-Khafaji,
2021; Kumar and Sahoo, 2021; Awasthi, 2017). Yet, there
has been relatively little research and discovery on the relevant
features involving bi-univalent function subclasses. However,
much attention has been focused on bi-univalent functions’ ini-
tial coefficients (Al-Ameedee et al., 2020; Rossdy et al., 2021;
Soni et al., 2018).

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Rossdy et. al. Science and Technology Indonesia, 8 (2023) 195-204

Gradshteyn and Ryzhik (2014) published a formulation
for the Bernoulli polynomials in 1980, which has substantial
uses in number theory and classical analysis. The Bernoulli
polynomials are featured in differentiable periodic functions
in the integral form of the functions because they are used for
polynomial approximation of these functions. The polyno-
mials are used to represent the remainder term of the Euler-
Maclaurin quadrature rule in its composite form as well. The
Bernoulli polynomials Bn(x) are commonly defined (Natalini
and Bernardini, 2003) using the generating function:

G(x, t) B
text

et − 1
=

∞∑︁
n=0

Bn(x)
n!

tn , |t| < 2𝜋,

where for each nonnegative integer n, Bn(x) are polynomials
in x.

Since

n−1∑︁
j=0

(
n
j

)
Bj(x) = nxn−1 , n = 2, 3, · · · ,

the Bernoulli polynomials can be calculated readily via
recursion.

The initial Bernoulli polynomials are

B0(x) = 1,

B1(x) = x − 12 ,

B2(x) = x2 − x + 16 ,

B3(x) + x3 − 32 x
2 + 12 x, · · ·

Moreover, Bernoulli numbers Bn B Bn(0) can be directly
generated by setting x = 0 in the Bernoulli polynomials. The
initial Bernoulli numbers are

B0(x) = 1,

B1(x) = − 12 ,

B2(x) = 16 ,

B4(x) = − 130

B2n+1 = 0, ∀n = 1, 2, · · ·

Furthermore, Bernoulli numbers Bn can be produced using
the Einstein function E(z):

E(z) B
z

ez − 1
=

∞∑︁
n=0

Bn
n!
zn.

The name of Einstein function is sometimes applied in
mathematics for one of the functions (see (Abramowitz and
Stegun, 1972; Weisstein, 2022)):

E1(z) B zez−1 ,

E2(z) B z
2ez

(ez−1)2 ,

E3(x) B log(1 − e−z),

E4(x) B zez−1 − log(1 − e
−z).

Both E1 and E2 exhibit these desirable properties. E1
and E2 (convex functions) have a symmetric range along the
real axis and starlike range about E1(0) = E2(0) = 1 and
ℝ(E1(z)) > 0, ℝ(E2(z)) > 0, ∀z ∈ 𝔻. The series representa-
tion is given by

E1(z) =1 +
∞∑︁
n=1

Bn
n!
zn

E2(z) =1 +
∞∑︁
n=1

(1 − n)Bn
n!

zn ,

where Bn denotes the nth Bernoulli number. However,
E′1(0) and E

′
2(0) ≯ 0, therefore we must establish new func-

tions E(z) B E1(z) + z and 𝔼(z) B E2(z) + 12 z. A significant
function class will be known as P and P defines the function
family 𝜙 that is restricted by the image domain of 𝜙 (𝜙 is a
convex function with Re (𝜙) > 0 in 𝔻) being symmetric along
the real axis and starlike about 𝜙(0) = 1 with 𝜙′(0) > 0. We
can now say that E, 𝔼 ∈ P. The following are the series repre-
sentations:

E(z) = 1 + z +
∞∑︁
n=1

Bn
n!
zn ,

and

𝔼(z) = 1 +
1
2
z +

∞∑︁
n=1

(1 − n)Bn
n!

zn ,

The contour integral (see (Arfken and Weber, 1999) ) can
be used to define the nth Bernoulli number, Bn:

Bn =
n!

2𝜋i

∮
z

ez − 1
dz
zn+1

,

where the radius of the contour encircling the origin is
less than 2𝜋i. El-Qadeem et al. (2022b) presented outcomes
relating to the first Einstein function E1, while El-Qadeem et al.
(2022a) worked on the second Einstein function E2.

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Rossdy et. al. Science and Technology Indonesia, 8 (2023) 195-204

Definition 1.3 Let
∑

indicate the bi-univalent function class in
𝔻. A function f ∈

∑
is said to be in the class N s,m,k

_ ,𝛼 (
∑

, E)
for 0 < _ < 1, 0 < 𝛼 < 1, m ∈ ℕ = {1, 2, · · · }, b ∈
ℂ\Z−0 , s ∈ ℂ, k ∈ ℕ0, if the subsequent subordination satisfies:

(1 − 𝛽)
Ds,m,k
_ ,𝛼 f (z)
z

+ 𝛽
(
Ds,m,k
_ ,𝛼 f (z)

)′
≺ E(z),

(1 − 𝛽)
Ds,m,k
_ ,𝛼 f (z)
w

+ 𝛽
(
Ds,m,k
_ ,𝛼 f (w)

)′
≺ E(w),

where Ds,m,k
_ ,𝛼 f (z) and g are denoted by (4) and (3), respec-

tively.

Definition 1.4 (Orloff, 2018) (Complex Logarithm Function)
The function log(z) is defined as

log(z) = log(|z|) + i arg(z),

where log |z| is the usual natural logarithm of a positive real
number.

Inspired by Sivasubramanian et al. (2014) , Rossdy et al.
(2022) , Zhang et al. (2021) , Saheb and Al-Khafaji (2021) ,
and El-Qadeem et al. (2022b) , we propose in this paper a
subclass of analytic bi-univalent function connected to the first
Einstein function, E(z). We obtain the covering theorem for
bi-univalent functions; the theorem states that each function’s
range in the class N s,m,k

_ ,𝛼 (
∑

, E) must encompass a disk with a
minimum radius of 14 . We also find the distortion theorem,
the growth theorem, and the convexity radius for functions in
the class N s,m,k

_ ,𝛼 (
∑

, E).

2. MAIN RESULTS

2.1 Covering Theorem
Firstly, we discover the covering theorem for the class N s,m,k

_ ,𝛼 (
∑

,
E) provided by the following:

Theorem 2.1.1 The range of each function of the class N s,m,k
_ ,𝛼

(
∑

, E) includes the disk {w ∈ ℂ : |w| < 14 }.

Proof. A disk automorphism is used to derive the function
f from a given function f ∈ N s,m,k

_ ,𝛼 (
∑

, E) and a fixed Z ∈ 𝔻
where

F(z) =
f
(
z+Z

1+Zz

)
− f (Z)

(1 − |Z |2) f ′(Z)
= z + A2(Z)z2 + · · · , forz ∈ 𝔻.

(5)

Then, we have

F(z) =z +


−6

(
2a2

(
1+b
2+b

)s
[1 + _ (1 − 𝛼)m]k(1 + 𝛽)Z+

2

(
6a2

(
1+b
2+b

)s
[1 + _ (1 − 𝛼)m]k(1 + 𝛽)

2a3
(

1+b
3+b

)s
[1 + 2_ (1 − 𝛼)m]k(1 + 2 𝛽)(3Z 2 − 1)

+12a3
(

1+b
3+b

)s
[1 + 2_ (1 − 𝛼)m]k(1 + 2 𝛽)Z + 5c21 Z

−3Z 2 + 1 + c21 (5 − 15Z
2) + 6c1Z + · · ·

)
−3c1 − 6Z

) + · · ·


z2 + · · · , z ∈ 𝔻

By performing a straightforward calculation, we get

F(z) =

z +
[ (1 − 3Z 2)(2a2a3 ( 1+b2+b )s[1 + _ (1 − 𝛼)m]k(1 + 2 𝛽)

2

(
a2

(
1+b
2+b

)s
[1 + _ (1 − 𝛼)m]k(1 + 𝛽) + 2a3

(
1+b
3+b

)s
5c21
6 − 1 + · · ·

)
[1 + 2_ (1 − 𝛼)m]k(1 + 2 𝛽)Z + 56c

2
1 Z −

c1
2 − Z + · · ·

)

− Z +
1
w

]
z2 + · · · ,

z ∈ 𝔻, is analytic and bi-univalent in 𝔻. Then, by incorporating

© 2023 The Authors. Page 198 of 204



Rossdy et. al. Science and Technology Indonesia, 8 (2023) 195-204

the inequality (4) with

[ (1 − 3Z 2)(2a2a3 ( 1+b2+b )s[1 + _ (1 − 𝛼)m]k(1 + 2 𝛽)
2

(
a2

(
1+b
2+b

)s
[1 + _ (1 − 𝛼)m]k(1 + 𝛽) + 2a3

(
1+b
3+b

)s
5c21
6 − 1 + · · ·

)
[1 + 2_ (1 − 𝛼)m]k(1 + 2 𝛽)Z + 56c

2
1 Z −

c1
2 − Z + · · ·

)

− Z +
1
w

]
≤ 2,

we find that

|w| ≥
1

(1−3Z 2 )

(
2a2a3

(
1+b
2+b

) s
[1+_ (1−𝛼)m]k (1+2 𝛽 )

2

(
a2

(
1+b
2+b

) s
[1+_ (1−𝛼)m]k (1+𝛽 )+2a3

(
1+b
3+b

) s

5c2
1

6 −1+···

)
[1+2_ (1−𝛼)m]k (1+2 𝛽 )Z+ 56 c

2
1 Z −

c1
2 −Z+···

)

In view of Brange’s work (De Branges, 1985) we find that
|A2(Z)| ≤ 2, therefore

|w| ≥
1
4

2.2 Distortion and Rotation Theorems
The next theorem, which gives a vital estimate, is used to de-
velop the distortion theorem and accompanying findings:

Theorem 2.2.1 For N s,m,k
_ ,𝛼 (

∑
, E), we have:

����zf ′′(z)f ′(z) − 2r21 − 3r2
���� ≤ 4r1 − 3r2 , |z| = r < 1. (6)

Proof. A disk automorphism defined in (5) is used to de-
rive the function F for a given function f ∈ N s,m,k

_ ,𝛼 (
∑

, E) and
a fixed Z ∈ 𝔻. Hence, using the elementary transformation,
we get f ∈ N s,m,k

_ ,𝛼 (
∑

, E), and a simple computation gives us

A2(Z) =
(1 − 3|Z |2)( f ′′(Z))

2( f ′(Z))
− Z̄ . (7)

Moreover, following the lead from Brange’s work (De Branges,
1985) , we can deduce that |A2(Z)| ≤ 2|. As a result, we may
get the inequality (6) by using the bound A2(Z) in (7) and
replacing Z with z.

After establishing Theorem 2.2.1, the following distortion
theorem can be shown:

Theorem 2.2.2 For each f ∈ N s,m,k
_ ,𝛼 (

∑
, E), we have:

(1 −
√

3r)
2
√

3−1
3

(1 +
√

3r)
2
√

3+1
3

< | f ′(z)| ≤
(1 +

√
3r)

2
√

3+1
3

(1 −
√

3r)
2
√

3−1
3

, |z| = r < 1.

(8)

Proof. From inequality (6), we obtain

−
4r

1 − 3r2
<
zf ′′(z)
f ′(z)

−
2r2

1 − 3r2
≤

4r
1 − 3r2

, |z| = r < 1.

(9)

By taking the real component of (8), we get

2r2

1 − 3r2
−

4r
1 − 3r2

< Re

{
zf ′′(z)
f ′(z)

}
<

2r2

1 − 3r2

+
4r

1 − 3r2
, |z| = r < 1. (10)

Since f ′(z)| ≠ 0 and f ′(0) = 1, it is possible to allocate a branch
of log f ′(z) that has a single value and which disappears at the
origin. As a result of utilising logarithmic differentiation and
Definition 1.4, we can find that

Re

{
zf ′′(z)
f ′(z)

}
=z Re

{
zf ′′(z)
f ′(z)

}

= r
𝜕

𝜕r
Re{log | f ′(z)|}, z = rei\ .

© 2023 The Authors. Page 199 of 204



Rossdy et. al. Science and Technology Indonesia, 8 (2023) 195-204

We then employ the above identity in (10) and obtain

2r − 4
1 − 3r2

<
𝜕

𝜕r
log | f ′(z)| <

2r + 4
1 − 3r2

, z = rei\ . (11)

With \ as a constant, we integrate the inequality (11) from
0 to R with respect to r which yields the following expression:

− 2
R∫
0

r − 2
3r2 − 1

dr <

R∫
0

𝜕

𝜕r
log | f ′(rei\)|𝜕r

< −2
R∫
0

r + 2
3r2 − 1

dr.

By employing the partial fractions, we obtain

− 2
R∫
0

6 +
√

3

2
√

3(3r +
√

3)
+ f rac6 −

√
32

√
3(3r −

√
3)dr

<

R∫
0

𝜕

𝜕r
log | f ′(rei\)|𝜕r < −2

R∫
0

−6 +
√

3

2
√

3(3r +
√

3)

+
6 +

√
3

2
√

3(3r −
√

3)
dr.

Using the technique of substitution, we get

[(
2
√

3 − 1
3

)
log(

√
3 − 3r) −

(
1 + 2

√
3

3

)
log(

√
3 + 3r)

]R
0

< log | f ′(rei\)| <
[(

2
√

3 − 1
3

)
log(

√
3 + 3r)

−
(
1 + 2

√
3

3

)
log(

√
3 − 3r)

]R
0

Then, by using the logarithmic quotient rule, we have

log

[
(1 −

√
3R)

2
√

3−1
3

(1 +
√

3R)
2
√

3+1
3

]
< log | f ′(rei\)|

≤
[
(1 +

√
3R)

2
√

3−1
3

(1 −
√

3R)
2
√

3+1
3

]
(12)

Finally, we attain (8) by exponentiating (12).

As a result, we intend to point out the fact that the upper
and lower bounds of the distortion factor | f ′(z)| for the class
N s,m,k

_ ,𝛼 (
∑

, E) are obtained by essentially putting into consider-
ation the real component of inequality (6) in Theorem 2.2.1.
However, by considering the imaginary part, we may get a con-
dition for the rotation factor |arg f ′(z)|. Hence, the theorem
of rotation is as follows:

Theorem 2.2.3 For each f ∈ N s,m,k
_ ,𝛼 (

∑
, E), we have:

|arg f ′(z)| ≤
2
√

3
log

[ 1 + √3r
1 −

√
3r

]
, |z| = r < 1.

Proof. From inequality (9), we attain

2r2

1 − 3r2
−

4r
1 − 3r2

<
zf ′′(z)
f ′(z)

≤
2r2

1 − 3r2
+

4r
1 − 3r2

,

|z| = r < 1. (13)

By considering the imaginary component only from (13), we
get:

−4r
1 − 3r2

< lm
zf ′′(z)
f ′(z)

≤
4r

1 − 3r2
, |z| = r < 1. (14)

Since | f ′(z)| ≠ 0 and f (0) = 1, it is possible to allocate a branch
of log f ′(z) that has a single value and which disappears at the
origin. Thus, by employing the logarithmic differentiation
and Definition 1.4, we have

lm
zf ′′(z)
f ′(z)

= z
2f ′′(z)
f ′(z)

= r
𝜕

𝜕r
arg f ′(z), z = rei\ .

As a result of utilising the above inequality in (14), we obtain:

−4
1 − 3r2

<
𝜕

𝜕r
arg f ′(rei\) ≤

4
1 − 3r2

, z = rei\ . (15)

© 2023 The Authors. Page 200 of 204



Rossdy et. al. Science and Technology Indonesia, 8 (2023) 195-204

The desired result is obtained by integrating the inequality
(15) from 0 to R with respect to r by keeping \ constant.

2.3 Radius of Convexity
Another area where the inequality (6) is related is the radius of
convexity. The theorem below estimates the convexity radius
for functions in the class N s,m,k

_ ,𝛼 (
∑

, E):

Theorem 2.3.1 For every positive number, the function f ∈
N s,m,k

_ ,𝛼 (
∑

, E) maps the disk |z| < p into a convex domain such
that p <

√
5−2 ≈ 0.23607.

Proof. Based on inequality (6), we may evaluate:

�����zf ′′(z)f ′(z)
����� ≤ 2r21 − 3r2 + 4r1 − 3r2 , |z| = r < 1. (16)

Next, we have a double inequality derived from (16), as repre-
sented below:

2r2 − 4r
1 − 3r2

<
zf ′′(z)
f ′(z)

≤
2r2 + 4r
1 − 3r2

|z| = r < 1.

Subsequently, by using a simple computation, we get

1 − r2 − 4r
1 − 3r2

< 1 +
zf ′′(z)
f ′(z)

≤
1 − r2 + 4r

1 − 3r2
|z| = r < 1.

(17)

By taking the real value from (17), we acquire

Re

{
1 +

zf ′′(z)
f ′(z)

}
>

1 − r2 − 4r
1 − 3r2

, |z| = r < 1.

but 1−r
2 −4r

1−3r2 > 0 for r <
√

5 − 2 ≈ 0.23607, and thence f
maps such a disk |z|, r onto a convex domain. Accordingly, this
demonstrates our result.

2.4 Growth Theorem
The distortion result from Theorem 2.2.2 can be used to derive
the lower and upper bounds of f ∈ N s,m,k

_ ,𝛼 (
∑

, E). We can
subsequently prove the growth theorem as follows:

Theorem 2.4.1 For each f ∈ N s,m,k
_ ,𝛼 (

∑
, E), we have

−
√

3

2
2(2+

√
3)

3

[
(1 −

√
3r)

2(1+
√

3)
3 2F1

(
1 + 2

√
3

3
,

2 + 2
√

3
3

;

5 + 2
√

3
3

;
1 −

√
3r

2

)
− 2F1

(
1 + 2

√
3

3
,

2 + 2
√

3
3

;

5 + 2
√

3
3

;
1
2

)]
< | f (z)| ≤ 2

(
2
√

3−7
3

)
(3 +

√
3) (18)

[
(1 −

√
3r)

(
2−2

√
3

3

)
2F1

(
1 − 2

√
3

3
,

2 − 2
√

3
3

;
5 − 2

√
3

3
;

1 −
√

3r
2

)
− 2F1

(
1 − 2

√
3

3
,

2 − 2
√

3
3

;
5 − 2

√
3

3
;
1
2

)]
,

|z| = r < 1.

Proof. Let f ∈ N s,m,k
_ ,𝛼 (

∑
, E) and z = rei\ , where 0 < r < 1.

We integrate the inequality from 0 to R with respect to r using
Theorem 2.2.2,

R∫
0

(1 −
√

3r)
2
√

3−1
3

(1 +
√

3r)
2
√

3+1
3

dr <

R∫
0

| f ′(z)|𝜕r ≤

R∫
0

(1 +
√

3r)
2
√

3−1
3

(1 −
√

3r)
2
√

3+1
3

dr.

Next, we obtain


−
√

3(1 −
√

3r)
2(1+

√
3)

3 2F1

(
1+2

√
3

3 ,
2+2

√
3

3 ;
5+2

√
3

3 ;
1−

√
3r

2

)
2

2(2+
√

3)
3



R

0

<
[
| f (rei\)|

]R
0 ≤

(19)

© 2023 The Authors. Page 201 of 204



Rossdy et. al. Science and Technology Indonesia, 8 (2023) 195-204


2

2
3 (

√
3−2)√3(1 −

√
3r)

(
2−2

√
3

3

)
2F1

(
1−2

√
3

3 ,
2−2

√
3

3 ;
5−2

√
3

3 ;

√
3 − 1

1 −
√

3r2

) 

R

0

.

Thus, a simple computation from (19) results in the double in-
equality (18).

The growth and distortion theorems can be used to achieve
the following inequality:

Theorem 2.4.2 For each f ∈ N s,m,k
_ ,𝛼 (

∑
, E), we have

r

(
(1−3

√
3r)

2
√

3−1
3

(1+3
√

3r)
2
√

3+1
3

)
−

√
3

2
2(2+

√
3)

3

[
(1 −

√
3r)

2(1+
√

3)
3 2F1

(
1+2

√
3

3 ,
2+2

√
3

3 ;
5+2

√
3

3 ;

1−
√

3r
2

)
− 2F1

(
1+2

√
3

3 ,
2+2

√
3

3 ;
5+2

√
3

3 ;
1
2

)] < ���zf ′(z)f (z) ��� ≤

r

(
(1+3

√
3r)

2
√

3−1
3

(1−3
√

3r)
2
√

3+1
3

)
2

(
2
√

3−7
3

)
(3+

√
3)

[
(1 −

√
3r)

2(1−
√

3)
3 2F1

(
1−2

√
3

3 ,
2−2

√
3

3 ;

5−2
√

3
3 ;

1−
√

3r
2

)
− 2F1

(
1−2

√
3

3 ,
2−2

√
3

3 ;
5−2

√
3

3 ;
1
2

)] ,

0 < |z| = r < 1.

Proof. By utilising Theorem 2.4.1 and Theorem 2.2.2, we get

(
(1−3

√
3r)

2
√

3−1
3

(1+3
√

3r)
2
√

3+1
3

)
−

√
3

2
2(2+

√
3)

3

[
(1 −

√
3r)

2(1+
√

3)
3 2F1

(
1+2

√
3

3 ,
2+2

√
3

3 ;
5+2

√
3

3 ;

1−
√

3r
2

)
− 2F1

(
1+2

√
3

3 ,
2+2

√
3

3 ;
5+2

√
3

3 ;
1
2

)] < ��� f ′(z)f (z) ��� ≤
(20)(

(1+3
√

3r)
2
√

3−1
3

(1−3
√

3r)
2
√

3+1
3

)
2

(
2
√

3−7
3

)
(3+

√
3)

[
(1 −

√
3r)

2(1−
√

3)
3 2F1

(
1−2

√
3

3 ,
2−2

√
3

3 ;

5−2
√

3
3 ;

1−
√

3r
2

)
− 2F1

(
1−2

√
3

3 ,
2−2

√
3

3 ;
5−2

√
3

3 ;
1
2

)] ,

|z| = r < 1.

Hence, from (20), by considering zf
′ (z)
f (z) , we obtain

r

(
(1−3

√
3r)

2
√

3−1
3

(1+3
√

3r)
2
√

3+1
3

)
−

√
3

2
2(2+

√
3)

3

[
(1 −

√
3r)

2(1+
√

3)
3 2F1

(
1+2

√
3

3 ,
2+2

√
3

3 ;
5+2

√
3

3 ;

1−
√

3r
2

)
− 2F1

(
1+2

√
3

3 ,
2+2

√
3

3 ;
5+2

√
3

3 ;
1
2

)] < ���zf ′(z)f (z) ��� ≤

r

(
(1+3

√
3r)

2
√

3−1
3

(1−3
√

3r)
2
√

3+1
3

)
2

(
2
√

3−7
3

)
(3+

√
3)

[
(1 −

√
3r)

2(1−
√

3)
3 2F1

(
1−2

√
3

3 ,
2−2

√
3

3 ;

5−2
√

3
3 ;

1−
√

3r
2

)
− 2F1

(
1−2

√
3

3 ,
2−2

√
3

3 ;
5−2

√
3

3 ;
1
2

)] ,

© 2023 The Authors. Page 202 of 204



Rossdy et. al. Science and Technology Indonesia, 8 (2023) 195-204

|z| = r < 1.

Therefore, the desired result is achieved. The combination of
growth and distortion theorems produces a useful inequality
where the starlikeness properties can be determined.

3. CONCLUSION

Our motivation comes from the aspiration to find numerous
novel and useful applications for the new generalised operator
Ds,m,k
_ ,𝛼 f (z) proposed by Rossdy et al. (2022) . Therefore, in

this paper, we found the theorems of covering, rotation, dis-
tortion, growth and the convexity radius for functions of the
class N s,m,k

_ ,𝛼 (
∑

, E) of bi-univalent functions connected with the
Einstein function and Ds,m,k

_ ,𝛼 f (z) by using the subordination
method.

4. ACKNOWLEDGMENT

We gratefully acknowledge the insightful comments and thor-
ough reading of the manuscript by the reviewers.

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© 2023 The Authors. Page 204 of 204


	INTRODUCTION
	MAIN RESULTS
	Covering Theorem
	Distortion and Rotation Theorems
	Radius of Convexity
	Growth Theorem

	CONCLUSION
	ACKNOWLEDGMENT