Journal-18-CKMS37-1-2022-113-123
                     Commun. Korean Math. Soc. 37 (2022), No. 1, pp. 113–123
https://doi.org/10.4134/CKMS.c200468
pISSN: 1225-1763 / eISSN: 2234-3024
SHARP COEFFICIENT INEQUALITIES FOR CERTAIN
SUBCLASSES OF BI-UNIVALENT BAZILEVIČ FUNCTIONS
Amol Bhausaheb Patil
Abstract. In the present paper, we introduce the subclasses B1Σ(µ),
B1Σ(µ, γ) and UΣ(µ, γ) of bi-univalent Bazilevič functions which are de-
fined in the open unit disk D. Further, we obtain sharp estimates on
initial coefficients a2, a3, a4 and also sharp estimate on the Fekete-Szegö
functional a3 − ka
2
2 for the functions belong to these subclasses.
1. Introduction
Let A denote the class of analytic functions of the form:
∑∞
(1) f(z) = z + a nnz , (z ∈ D, n ∈ N, an ∈ C) ,
n=2
which are defined in the open unit disk D := {z ∈ C : |z| < 1} and satisfies
′
the standard normalization conditions [f(z) = 0, f (z) = 1]z=0. Also, let S
represent the subclass of A, that contain functions of the form (1) which are
2
univalent in D. The Koebe function z/(1− z) = z + 2z2 + 3z3 + · · · is the
most important member of the class S. Further, let S∗ denote the subclass of
S, that contain functions which are star-like in D. Whereas, f ∈ A given by
(1) is known as a star-like function if it maps the open unit disk D to a star-like
domain with respect to the origin. In addition, if f ∈ S∗, then we have:
{ }
′
zf (z)
< > 0, (z ∈ D) .
f(z)
It is well known that if f ∈ S∗, then |an| ≤ n for every n > 1, (n ∈ N) and the
result is sharp for the Koebe function.
According to the Koebe one-quarter theorem (see [7]), the image of D under
every f ∈ S contains a disk of radius one-quarter centered at origin. Thus, every
f ∈ S has an inverse f−1 : f(D) → D that satisfies f−1(f(z)) = z; (|z| < 1)
and f(f−1(w)) = w; (|w| < r0(f), r0(f) ≥ 1/4). Moreover, for f ∈ S, an
Received December 16, 2020; Accepted August 13, 2021.
2010 Mathematics Subject Classification. Primary 30C45, 30C50.
Key words and phrases. Analytic function, univalent function, bi-univalent function, star-
like function, Bazilevič function.
©c 2022 Korean Mathematical Society
113
114 A. B. PATIL
analytic, univalent continuation of the inverse function f−1 ≡ g to D is of the
form:
(2) g(w) = w + (−a2)w
2 + (2a2 32 − a3)w + (5a2a3 − 5a
3 4
2 − a4)w + · · · .
A function f ∈ S given by (1), is said to be bi-univalent if f−1 ∈ S and the
class of all such functions denoted by Σ is said to be the bi-univalent function
class. The functions z/ (1− z), − log (1− z) and (1/2) log [(1 + z) / (1− z)] are
( ) ( )
the members of the class Σ. However, the functions z− z2/2 , z/ 1− z2 and
also the Koebe function are not the members of Σ.
Lewin [12] introduced the concept of class Σ and proved that |a2| < 1.51
for functions in it. After which, Brannan and Clunie [5] proved that |a2|√ f∈Σ
≤
2. Later, Netanyahu [15] showed that max |a2|f∈Σ = 4/3, whereas Styer and
Wright [24] showed the existence of f ∈ Σ for which |a2| > 4/3. Further, Tan
[25] proved that |a2| ≤ 1.485 for functions in Σ. After invention of the class
Σ, many researchers have been working to find out the connection between the
coefficient bounds and geometrical properties of the functions in it.
Indeed, Lewin [12], Brannan and Taha [6], Srivastava et al. [23] etc. provided
a solid base for the study of bi-univalent functions. After which, many re-
searchers viz. [9,10,18,20,22] (also see the references therein) introduced several
subclasses of Σ and found estimates on initial coefficients for functions in them.
However, still the problem of sharp coefficient bound for |an|, (n = 3, 4, 5, . . .)
is open.
Ram Singh [21] introduced the class B1(µ) of Bazilevič functions, that consist
of functions f ∈ A for which:
{[ ] }µ−1
f(z) ′
< f (z) > 0, (z ∈ D, µ ≥ 0) .
z
In fact, it is known (see [13]) that B1(µ) ⊂ S and B (0) ≡ S
∗
1 . Moreover, the
′
subclass B1(1) satisfies the condition  0, z ∈ D and reduce to the
close-to-convex class. Singh [21] and Ali et al. [3] have used two different ways
to obtain sharp bounds for first three coefficients of the class B1(µ).
We need the following lemmas to prove our main results.
Lemma 1.1 ([7]). Let P denote the class of all analytic functions in D with
positive real part. For z ∈ D if
∑∞
P (z) = 1 + c nnz ∈ P,
n=1
then |cn| ≤ 2 for each n ∈ N.
Lemma 1.2 ([14]). If the functions F1 and F2 defined by
∑∞ ∑∞
F1 = 1 + bnz
n and F2 = 1 + c z
n
n belong to the class P,
n=1 n=1
SHARP COEFFICIENT INEQUALITIES FOR BI-UNIVALENT FUNCTIONS 115
∑∞
then the function F = 1 + 1 nn=1 bncnz also belong to P.2
Lemma 1.3 ([14]). Let h(z) = 1 + α1z + α z
2
2 + · · · and t(z) = 1 + t1(z) =
1 + β1z + β2z
2 + · · · be the functions of the class P and for m, k ∈ N set
[ ]
∑m
( )
1 1 m
(3) ηm = 1 + αk , η0 = 1.
2m 2 k
k=1
If An is defined by
∑∞ ∑∞
(4) A zn
n+1
n = (−1) η
n
n−1t1 (z),
n=1 n=1
then for each n ∈ N,
|An| ≤ 2.
In this paper, we define the subclasses B1Σ(µ), B1Σ(µ, γ) and UΣ(µ, γ) of
Σ that are associated with the Bazilevič functions (for more details about
Bazilevič functions see [4, 26, 27]). Moreover, by using the method of Ram
Singh [21] along with the equating coefficient trick of Srivastava et al. [23], we
obtain sharp bounds for the coefficients a2, a3 and a4 for the functions belong
to these subclasses.
2. Coefficient estimates for the class B1Σ(µ)
Definition. A function f(z) ∈ Σ of the form (1) is said to be in the class
B1Σ(µ); (µ > 0) if the following two conditions are fulfilled:
{ }
′ µ−1
zf (z)f(z)
< > 0, (z ∈ D)
zµ
and { }
′ µ−1
wg (w)g(w)
< > 0, (w ∈ D) ,
wµ
where g is of the form (2), be an extension of f−1 to D.
Theorem 2.1. Let f(z) ∈ B1Σ(µ), (µ > 0) be given by (1). Then we have the
following sharp estimates:
√
 2(3+µ) , 0 < µ ≤ 1,
√ (2+µ)(1+µ)
2
(5) |a2| ≤
 2 , µ ≥ 1,
2+µ
{
2(3+µ)
, 0 < µ ≤ 1,
(6) |a | ≤ (2+µ)(1+µ)
2
3 2 , µ ≥ 1,
2+µ

 2 4(1−µ)(5+3µ+µ
2)
+ 3 , 0 < µ ≤ 1,
(7) |a4| ≤ 3+µ 3(2+µ)(1+µ)
 2 , µ ≥ 1.
3+µ
116 A. B. PATIL
Proof. Since f ∈ B1Σ(µ), by definition we have
z1−µ
′
f (z)
(8) = P (z),
1−µ
f(z)
′
w1−µg (w)
(9) = Q(w),
1−µ
g(w)
for some P (z), Q(w) ∈ P. On setting
(10) P (z) = 1 + c1z + c2z
2 + · · ·+ cnz
n + · · · ,
(11) Q(w) = 1 + d1w + d2w
2 + · · ·+ d nnw + · · ·
and then comparing the coefficients in (8) and (9) we obtain
(12) (1 + µ) a2 = c1,
µ
(13) (2 + µ) a3 = c2 + (1− µ) c1a − (1− µ) a
2
2
2 2
,
[ µ ]
(14) (3 + µ) a4 = c3 + (1− µ) c2a2 + (1− µ) a3 − (1− µ) a
2 c1
2 2
µ (1− µ) (1 + µ)
− µ (1− µ) a 32a3 + a2,6
and
(15) − (1 + µ) a2 = d1,
( ) µ
(16) (2 + µ) 2a22 − a3 = d2 − (1− µ) d1a2 − (1− µ) a
2,
2 2
( )
(17) − (3 + µ) 5a32 − 5a2a3 + a4 = d3 − (1− µ) d2a2
[ µ ]
− (1− µ) a 23 + (1− µ) a2 d12
− 2 (1− µ) a2a3
[ ]
µ (1− µ) (1 + µ)
− 2 (1− µ) + a3
6 2
.
Equation (12) and (15) together yields
2
c1 = −d1 and |a2| ≤ .
1 + µ
Adding equations (13) and (16) we get
2 (c2 + d2) (1− µ)(18) a2 = + c
2
2 (2 + µ) 2 1
,
2 (1 + µ)
whereas, subtracting (16) from (13) and then using (18), we get
c2 (1− µ)
(19) a = + c23 .
(2 + µ) 2 12 (1 + µ)
SHARP COEFFICIENT INEQUALITIES FOR BI-UNIVALENT FUNCTIONS 117
Application of the fact |cn| ≤ 2, |dn| ≤ 2; n := 1, 2, . . . in (18) and (19) along
with 0 < µ ≤ 1 proves the first part of the inequalities (5) and (6).
To prove the second part, we use
( )
1 1
(20) c2 = c
2
1 + δ 2− |c1|
2 , |δ| ≤ 1,
2 2
which is a consequence of the Carathéodory-Toeplitz inequality:
∣ ∣
∣ 1 ∣ 1
∣c2 − c
2∣ 2
∣ 2 1
≤ 2− |c
∣ 1
| .
2
Performing elementary calculations along with the equality (20), we obtain the
second part of the inequalities (5) and (6).
Observe that the first parts of the inequalities for a2 and a3 are sharp for
the functions f1 and g1 defined by:
z1−µ
′ ′
f1(z) 1 + z w
1−µg1(w) 1− w= and = ,
1−µ
f1(z) 1− z
1−µ
g1(w) 1 + w
whereas the second parts are sharp for f2 and g2 that satisfies:
z1−µ
′
f (z) 1 + z2 w1−µ
′
2 g (w) 1 + w
2
= and 2 = .
1−µ 2 1−µ 2
f2(z) 1− z g2(w) 1− w
Next, for proof of the third inequality we compile the outputs of addition and
subtraction of equations (14) and (17), which yields
[ µ ]
(3 + µ) a = c 24 3 + (1− µ) c2a2 + (1− µ) a3 − (1− µ) a c1
2 2
µ (1− µ) (1 + µ)
− µ (1− µ) a a + a32 3 2,6
which, by eliminating a2 and a3 produce
[ ]
(1− µ) (3 + µ) c1c2 (1− 2µ)
(21) (3 + µ) a4 = c3 + + c
3 .
(1 + µ) 2 + µ 2 16 (1 + µ)
From this, the first part of the inequality (7) is trivial for 0 < µ ≤ 1/2. Next,
for 1/2 < µ ≤ 1 we use the equality (20) in (21) to eliminate c2, which on
applying simple calculus implies that the expression in the square bracket of
equation (21) attains its maximum when c1 = c2 = 2. Thus, for 1/2 < µ ≤ 1,
|a4| attain its maximum when |cn| = 2, (n := 1, 2, 3) which completes the first
part of the inequality (7).
Finally, to obtain bound on a4 for µ ≥ 1 we use Lemmas 1.2 and 1.3 given
by Nehari and Netanyahu [14]. From the equation (4) we get
3
(22) A3 = β3 − 2η1β1β2 + η2β1 .
Lemma 1.2 along with |An| ≤ 2 and P (z) = 1 + c1z + c
2
2z + · · · ∈ P yields
∣ ∣
∣1 1 1 ∣3
(23) ∣ β c 3∣
∣ 3 3
− η1β1β2c1c2 + η2β1 c1 ≤ 2.
2 2 8 ∣
118 A. B. PATIL
On comparing (21) and (23) with µ ≥ 1, we conclude that
∣ [ ]∣
∣ ∣
∣ (1− µ) (3 + µ) c1c2 (1− 2µ) ∣
(3 + µ) |a4| = ∣c3 + + c
3
∣ ≤ 2,
∣ (1 + µ) 2 + µ 2 16 (1 + µ) ∣
if there exist the functions h(z), t(z) ∈ P given by
h(z) = 1 + α z + α z2 + α z31 2 3 + · · · and t(z) = 1 + β1z + β2z
2 + β3z
3 + · · ·
such that β3 = 2,
1 (µ− 1) (3 + µ) 1 3 (µ− 1) (2µ− 1) (3 + µ)
(24) η1β1β2 = and η2β1 = ,
2 (1 + µ) (2 + µ) 8 36 (1 + µ)
where ηm, (m ∈ N) is given by the equation (3), which implies that
( ) ( )
1 1 1 1
(25) η1 = 1 + α1 and η2 = 1 + α1 + α2 .
2 2 4 2
Also, on choosing β1 = β2 = 2, relations in (24) gives
(µ− 1) (3 + µ) (µ− 1) (2µ− 1) (3 + µ)
(26) η1 = and η2 = .
2 (1 + µ) (2 + µ) 36 (1 + µ)
Equating the values of η1 in (25) and (26), we get
−2 (5 + µ)
(27) α1 = .
(1 + µ) (2 + µ)
Since |α1| ≤ 2 for all µ ≥ 1, this value is acceptable. Next, on equating the
values of η2 in (25) and (26) in light of (27), we obtain
( )
2 µ4 + 5µ3 + 11µ2 − 19µ+ 36
(28) α2 = ,3
3 (2 + µ) (1 + µ)
which also satisfies that |α2| ≤ 2 for all µ ≥ 1. Finally, to construct the
functions h(z) and t(z), it is evident to have t(z) = (1 + z) / (1− z) ∈ P and a
suitable choice of h(z) is the function
( )
L (1− z) M 1 +Nz2
h(z) = + ,
(1 + z) (1−Nz2)
where
(µ+ 5) (µ− 1) (µ+ 3)
L = , M =
(µ+ 1) (µ+ 2) (µ+ 1) (µ+ 2)
and ( )
µ4 + 2µ3 − 10µ2 − 14µ+ 21
N = .
2
3 (µ+ 1) (µ− 1) (µ+ 3)
Observe here that L,M,N are all positive, L+M = 1 and N ≤ 1 for µ ≥ 1 and
hence we have h(z) ∈ P. Moreover, on expanding h(z) in ascending powers of
z shows that the coefficients of z and z2 in this expansion are equal to α1 and
α2 given by equations (27) and (28), respectively. Hence, we have
(3 + µ) |a4| ≤ 2 for µ ≥ 1,
SHARP COEFFICIENT INEQUALITIES FOR BI-UNIVALENT FUNCTIONS 119
which proves the second part of the inequality (7) and the result is sharp for
the functions f3 and g3 defined by:
z1−µ
′
f (z) 1 + z3 w1−µ
′
g (w) 1 + w33 = and 3 = .
1−µ 3 1−µ 3
f3(z) 1− z g3(w) 1− w 
On setting µ = 1 in Theorem 2.1, we get the following corollary as an
improvement in Theorem 2 given by Srivastava et al. [23].
Corollary 2.2. Let f(z) ∈ HΣ(0) ≡ HΣ be given by (1). Then we have the
following sharp estimates:
√
2 2 1
|a2| ≤ , |a3| ≤ , |a4| ≤ .
3 3 2
Further, as a consequence of Theorem 2.1, we obtain the following result
known as the Fekete-Szegö problem for the class B1Σ(µ).
Theorem 2.3. Let f(z) ∈ B1Σ(µ), (µ > 0) be given by (1). Then we have the
following sharp estimate:
{
2(3+µ)
|1− k|, 0 < µ ≤ 1,
|a − ka2| ≤ (2+µ)(1+µ)
2
3 2 2 |1− k|, µ ≥ 1,
2+µ
where k is some real number.
Proof. Using equations (18) and (19), we get
( ) ( )
2 c2 (1− µ) (c2 + d2) (1− µ)a3 − ka2 = + c
2
1 − k + c
2
(2 + µ) 22 (1 + µ) 2 (2 + µ)
2 1
2 (1 + µ)
2c2 − k (c2 + d2) (1− k) (1− µ)
= + c2.
2 (2 + µ) 2 12 (1 + µ)
Now, using the equality (20) along with the fact c1 = −d1 yields
 ( ) 
c2
2
1 + δ 4− |c1|
2 (1− µ)a  23 − ka2 = (1− k) + c , |δ| ≤ 1,2 (2 + µ) 2 12 (1 + µ)
which, in light of Lemma 1.1 gives
[ ]
∣ ∣ 2 2 (1− µ)
∣a3 − ka
2∣
2 ≤ |1− k| + .(2 + µ) 2(1 + µ)
This proves the required inequality according to the restrictions on µ. 
Remark 2.4. Clearly, for k = 0 Theorem 2.3 gives the sharp bound for |a3| and∣ ∣
for k = 1 it shows that ∣a3 − a
2∣
2 = 0.
120 A. B. PATIL
3. Coefficient estimates for the class B1Σ(µ, γ)
We have used the generalization of the univalance criterion appeared in the
paper of Aksentév [1] (also see [2] and the result by Ozaki and Nunokawa [17]).
According to it, for f(z) ∈ A, if
∣ ∣
∣
z1−µ
′ ∣
∣ f (z) ∣
∣ − 1∣ < 1, (z ∈ D, 0 < µ ≤ 1),
∣ 1−µf(z) ∣
then f(z) is univalent in D and hence f(z) ∈ S. Also, let T (µ, γ) denote the
class of functions f(z) ∈ A such that
∣ ∣
∣ 1−µ ′ ∣
∣z f (z) ∣
∣ − 1∣ < γ, (z ∈ D, 0 < γ ≤ 1),
∣ 1−µf(z) ∣
where T (µ, 1) = T (µ). Clearly, T (µ, γ) ⊂ T (µ) ⊂ S. Moreover, for f(z) ∈
T (µ, γ) (see Kuroki et al. [11]), we have
( )
z1−µ
′
f (z)
< > 1− γ, (z ∈ D).
1−µ
f(z)
Further, Ponnusamy [19] shown that for f ∈ A,
∣ ∣
∣ ( )µ−1 ∣
∣ ′ f(z) ∣
∣f (z) − 1∣ < γ, (z ∈ D, µ > 0, 0 < γ < 1)
∣ z ∣
is a condition of star-likeness in D.
Definition. A function f(z) ∈ Σ of the form (1) is said to be in the class
B1Σ(µ, γ); (0 < µ ≤ 1, 0 < γ ≤ 1) if the following two conditions are fulfilled:
∣ ∣
∣ 1−µ ′ ∣
∣z f (z) ∣
∣ − 11−µ ∣ < γ, (z ∈ D)∣ f(z) ∣
and
∣ ∣
∣
w1−µ
′ ∣
∣ g (w) ∣
∣ − 1∣ < γ, (w ∈ D) ,
∣ 1−µg(w) ∣
where g is of the form (2), be an extension of f−1 to D.
Theorem 3.1. Let f(z) ∈ B1Σ(µ, γ), (0 < µ ≤ 1, 0 < γ ≤ 1) be given by (1).
Then we have the following sharp estimates:
√
2γ 2 (1− µ) γ2
|a2| ≤ + ,
2 + µ 2(1 + µ)
2γ 2 (1− µ) γ2
|a3| ≤ + ,
2 + µ 2(1 + µ)
SHARP COEFFICIENT INEQUALITIES FOR BI-UNIVALENT FUNCTIONS 121
[ ]
2 2
2γ 4γ (1− µ) 3 (1 + µ) + γ (1− 2µ) (2 + µ)
|a4| ≤ +
3 + µ 33 (2 + µ) (1 + µ)
and for some real number k,
[ ]
∣ ∣ 2γ 2 (1− µ) γ2
∣a 2∣3 − ka2 ≤ |1− k| + .2 + µ 2(1 + µ)
Proof. Given f ∈ B1Σ(µ, γ). Thus, we have
{ }
z1−µ
′
f (z)
< > 1− γ
1−µ
f(z)
and { }
w1−µ
′
g (w)
< > 1− γ.
1−µ
g(w)
Hence we can write
z1−µ
′
f (z)
(29) = (1− γ) + γ P (z)
1−µ
f(z)
and
w1−µ
′
g (w)
(30) = (1− γ) + γ Q(w),
1−µ
g(w)
where P (z), Q(w) ∈ P are of the form (10) and (11), respectively.
On equating the coefficients in (29) and (30), we obtain the similar equations
as the equations (12) to (17) with the replacement of c1, c2, c3 by c1γ, c2γ, c3γ
respectively. Hence, we can construct the further proof similarly as the proof
of Theorem 2.1. 
Observe that the classes B1Σ(µ); (0 < µ ≤ 1) and B1Σ(µ, γ); (γ = 1) are
agree with all their corresponding estimates of a2, a3 and a4.
Now we consider the class U(µ, γ), introduced by Obradović [16] that consists
of functions f ∈ A which satisfy the condition:
∣ ∣
∣( )1+µ ∣
∣ z ′ ∣
∣ f (z)− 1∣ < γ, (z ∈ U, 0 < µ < 1, 0 < γ < 1) .
∣ f(z) ∣
The univalency problem for this class U(µ, γ) with µ as a complex number has
been studied by Fournier and Ponnusamy [8]. Observe that, for µ < 0 this
class correlates to the class B1Σ(µ, γ).
Definition. A function f(z) ∈ Σ of the form (1) is said to be in the class
UΣ(µ, γ); (0 < µ ≤ 1, 0 < γ ≤ 1) if the following two conditions are fulfilled:
∣ ∣
∣( )1+µ ∣
∣ z ′ ∣
∣ f (z)− 1∣ < γ, (z ∈ D)
∣ f(z) ∣
122 A. B. PATIL
and ∣ ∣
∣( )1+µ ∣
∣ w ′ ∣
∣ g (w)− 1∣ < γ, (w ∈ D) ,
∣ g(w) ∣
where g is of the form (2), be an extension of f−1 to D.
The following theorem, which we state without proof here, is a consequence
of Theorem 3.1.
Theorem 3.2. Let f(z) ∈ UΣ(µ, γ), (0 < µ ≤ 1, 0 < γ ≤ 1) be given by (1).
Then we have the following sharp estimates:
√
2γ 2 (1 + µ) γ2
|a2| ≤ + ,
2− µ 2(1− µ)
2γ 2 (1 + µ) γ2
|a3| ≤ + ,
2− µ 2(1− µ)
[ ]
2 2
2γ 4γ (1 + µ) 3 (1− µ) + γ (1 + 2µ) (2− µ)
|a4| ≤ +
3− µ 33 (2− µ) (1− µ)
and for some real number k,
[ ]
∣ ∣
2 2γ 2 (1 + µ) γ
2
∣a3 − ka ∣2 ≤ |1− k| + .2− µ 2(1− µ)
Remark 3.3. All the above results of the class UΣ(µ, γ) with −1 ≤ µ < 0 are
agree with the class B1Σ(µ, γ).
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Amol Bhausaheb Patil
Department of First Year Engineering
AISSMS College of Engineering
Pune 411001, India
Email address: [email protected]