Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 4 Issue 2, (2012), Pages 29-39 .
CERTAIN COEFFICIENT INEQUALITIES FOR SAKAGUCHI
TYPE FUNCTIONS AND APPLICATIONS TO FRACTIONAL
DERIVATIVES
(COMMUNICATED BY R.K. RAINA)
B. SRUTHA KEERTHI, V.G. SHANTHI, B. ADOLF STEPHEN
Abstract. In the present paper, sharp upper bounds of |a3 − µa22 | for the
functions f (z) = z +a2 z 2 +a3 z 3 +· · · belonging to a new subclass of Sakaguchi
type functions are obtained. Also, application of our results for subclass of
functions defined by convolution with a normalized analytic function are given.
In particular, Fekete-Szegö inequalities for certain classes of functions defined
through fractional derivatives are obtained.
1. Introduction
Let A be the class of analytic functions of the form
f (z) = z +
∞
∑
an z n
(1.1)
n=2
defined on ∆ := {z : z ∈ C and |z| < 1} and S be the subclass of A consisting of
univalent functions.
For two functions f, g ∈ A, we say that the function f (z) is subordinate to g(z)
in ∆ and write f ≺ g or f (z) ≺ g(z), if there exists an anlytic function ω(z) with
ω(0) = 0 and |ω(z)| < 1, (z ∈ ∆), such that f (z) = g(ω(z)),
(z ∈ ∆). In particular, if the function g is univalent in ∆, the above subordination is equivalent to f (0) = g(0) and f (∆) ⊂ g(∆).
A function f (z) ∈ A is said to be in the class M (α, λ, t) if it satisfies
{
[
]
[
]}
(1 − t)zf ′ (z)
(1 − t)(z 2 f ′′ (z) + zf ′ (z))
Re (1 − λ)
+λ
> α,
(1.2)
f (z) − f (zt)
zf ′ (z) − tzf ′ (zt)
|t| ≤ 1, t ̸= 1, 0 ≤ λ ≤ 1, for some α ∈ [0, 1) and for all z ∈ ∆.
2000 Mathematics Subject Classification. 30C45.
Key words and phrases. Sakaguchi functions, Analytic functions, Subordination, Fekete-Szegö
inequality, Fractional derivatives.
c
⃝2012
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted November 2, 2011. Published March 21, 2012.
29
30
B. SRUTHA KEERTHI, V.G. SHANTHI, B. ADOLF STEPHEN
For, λ = 0, the function f (z) ∈ A reduces to the Sakaguchi type class S ∗ (α, t)
which satisfies
]
[
(1 − t)zf ′ (z)
Re
> α,
f (z) − f (zt)
|t| ≤ 1, t ̸= 1, for some α ∈ [0, 1) and for all z ∈ ∆ was introduced and studied by
Owa et al. [6, 7].
For λ = 0, α = 0 and t = −1, we get the class S ∗ (0, −1) studied by Sakaguchi
[8]. A function f (z) ∈ S ∗ (α, −1) is called Sakaguchi function of order α.
For, λ = 1, the function f (z) ∈ A reduces to the class T (α, t) which satisfies
[
]
(1 − t)(z 2 f ′′ (z) + zf ′ (z))
Re
> α,
zf ′ (z) − tzf ′ (zt)
|t| ≤ 1, t ̸= 1, for some α ∈ [0, 1) and for all z ∈ ∆.
In this paper, we define the following class M (ϕ, λ, t) which is the
generalization of the class M (α, λ, t).
Definition 1.1. Let ϕ(z) = 1 + B1 z + B2 z 2 + · · · be univalent starlike function with
respect to 1 which maps the unit disk ∆ onto a region in the right half plane which
is symmetric with respect to the real axis, and let B1 > 0. The function f ∈ A is
in the class M (ϕ, λ, t) if
[
]
[
]
(1 − t)zf ′ (z)
(1 − t)(z 2 f ′′ (z) + zf ′ (z))
(1 − λ)
+λ
≺ ϕ(z),
(1.3)
f (z) − f (zt)
zf ′ (z) − tzf ′ (zt)
|t| ≤ 1, t ̸= 1, 0 ≤ λ ≤ 1.
For λ = 0 and λ = 1 in (1.3) we obtain the classes S ∗ (ϕ, t) and T (ϕ, t) respectively
which were studied by Goyal et al. [1].
In the present paper, we obtain the Fekete-Szegö inequality for the functions in
the subclass M (ϕ, λ, t). We also give application of our results to certain functions
defined through convolution (or Hadamard product) and in particular, we consider
the class M γ (ϕ, λ, t) defined by fractional derivatives.
To prove our main results, we need the following lemmas:
Lemma 1.2. [3] If p1 (z) = 1 + c1 z + c2 z 2 + · · ·
real part in ∆, then


−4v + 2
2
|c2 − vc1 | ≤ 2


4v − 2
is an analytic function with positive
if v ≤ 0,
if 0 ≤ v ≤ 1,
if v ≥ 1.
When v < 0 or v > 1, the equality holds if and only if p1 (z) is (1 + z)/(1 − z)
or one of its rotations. If 0 < v < 1, then equality holds if and only if p1 (z) is
(1 + z 2 )/(1 − z 2 ) or one of its rotations. If v = 0, the equality holds if and only if
(
)
(
)
1 1
1+z
1 1
1−z
p1 (z) =
+ γ
+
− γ
(0 ≤ γ ≤ 1)
2 2
1−z
2 2
1+z
or one of its rotations. If v = 1, the equality holds if and only if p1 (z) is the
reciprocal of one of the functions such that the equality holds in the case of v = 0.
Also the above upper bound is sharp and it can be improved as follows when
0 < v < 1:
|c2 − vc21 | + v|c1 |2 ≤ 2 (0 < v ≤ 1/2)
CERTAIN COEFFICIENT INEQUALITIES FOR SAKAGUCHI TYPE FUNCTIONS...
31
and
|c2 − vc21 | + (1 − v)|c1 |2 ≤ 2 (1/2 < v ≤ 1).
Lemma 1.3. [2] If p(z) = 1 + c1 z + c2 z 2 + · · · is a function with positive real part,
then for any complex number µ,
|c2 − µc21 | ≤ 2 max{1, |2µ − 1|}
and the result is sharp for the functions given by
p(z) =
1 + z2
,
1 − z2
p(z) =
1+z
.
1−z
2. Main Results
Our main result is the following:
Theorem 2.1. If f (z) given by (1.1) belongs to M (ϕ, λ, t), then
|a3 − µa22 |
[
(
)
(
)
]

1+t (1+3λ)
1
2 2+t (1+2λ)

B2 + B12 1−t
2 − µB1
2

(1+2λ)(1−t)(2+t)
(1+λ)
1−t
(1+λ)




B1
≤ (1+2λ)(1−t)(2+t)



[
(
)
(
)
]


−
1
2 1+t (1+3λ)
2 2+t (1+2λ)
(1+2λ)(1−t)(2+t) B2 + B1 1−t (1+λ)2 − µB1 1−t (1+λ)2
if µ ≤ σ1 ,
if σ1 ≤ µ ≤ σ2 ,
if µ ≥ σ2
[
(
)
]
(1 − t)(1 + λ)2
B2
1 + t (1 + 3λ)
−1 +
+ B1
B1 (2 + t)(1 + 2λ)
B1
1 − t (1 + λ)2
and
[
(
)
]
(1 − t)(1 + λ)2
B2
1 + t (1 + 3λ)
σ2 :=
1+
+ B1
B1 (2 + t)(1 + 2λ)
B1
1 − t (1 + λ)2
The result is sharp.
where
σ1 :=
Proof. Let f ∈ M (ϕ, λ, t). Then there exists a Schwarz function ω(z) ∈ A such
that
[
]
[
]
(1 − t)zf ′ (z)
(1 − t)(z 2 f ′′ (z) + zf ′ (z))
(1 − λ)
+λ
= ϕ(ω(z))
(2.1)
f (z) − f (zt)
zf ′ (z) − tzf ′ (zt)
(z ∈ ∆; |t| ≤ 1, t ̸= 1).
If p1 (z) is analytic and has positive real part in ∆ and p1 (0) = 1, then
p1 (z) =
1 + ω(z)
= 1 + c1 z + c2 z 2 + · · · (z ∈ ∆).
1 − ω(z)
From (2.2), we obtain
c1
1
ω(z) = z +
2
2
Let
[
p(z) = (1 − λ)
(
)
c21
c2 −
z2 + · · ·
2
(2.2)
(2.3)
]
[
]
(1 − t)zf ′ (z)
(1 − t)(z 2 f ′′ (z) + zf ′ (z))
+λ
f (z) − f (zt)
zf ′ (z) − tzf ′ (zt)
= 1 + b1 z + b2 z 2 + · · · (z ∈ ∆),
(2.4)
32
B. SRUTHA KEERTHI, V.G. SHANTHI, B. ADOLF STEPHEN
which gives
b1 = a2 (1 − t)(1 + λ)
(2.5)
b2 = a22 (t2 − 1)(1 + 3λ) + a3 (2 − t − t2 )(1 + 2λ)
(2.6)
and
Since ϕ(z) is univalent and p ≺ ϕ, therefore using (2.3), we obtain
[ (
)
]
1
c21
1 2
B1 c1
z+
c2 −
B1 + c1 B2 z 2 + · · · (z ∈ ∆) (2.7)
p(z) = ϕ(ω(z)) = 1 +
2
2
2
4
Now from (2.4), (2.5), (2.6) and (2.7), we have,
a2 (1 − t)(1 + λ) =
B1 c1
,
2
a22 (t2 − 1)(1 + 3λ) + a3 (2 − t − t2 )(1 + 2λ)
(
)
1
c21
1
=
c2 −
B1 + c21 B2 , |t| ≤ 1, t ̸= 1
2
2
4
(2.8)
(2.9)
Therefore, we have
a3 − µa22 =
where
v :=
B1
[c2 − vc21 ]
2(1 + 2λ)(1 − t)(2 + t)
(2.10)
[
(
)
(
)
]
1
B2
1 + t (1 + 3λ)
2 + t (1 + 2λ)
1−
− B1
+
µB
1
2
B1
1 − t (1 + λ)2
1 − t (1 + λ)2
Our result now follows by an application of Lemma 1.2. To show that these bounds
are sharp, we define the functions Kδϕ (δ = 2, 3, . . . ) by
[
]
[
]
(1 − t)z(Kδϕ (z))′
(1 − t)(z 2 (Kδϕ (z))′′ + z(Kδϕ (z))′ )
(1 − λ)
+λ
Kδϕ (z) − Kδϕ (zt)
z(Kδϕ (z))′ − tz(Kδϕ (zt))′
= ϕ(z δ−1 ), Kδϕ (0) = 0 = (Kδϕ (0))′ − 1
and the function Fγ and Gγ (0 ≤ γ ≤ 1) by
[
]
[
]
(1 − t)z(Fγ (z))′
(1 − t)(z 2 (Fγ (z))′′ + z(Fγ (z))′ )
(1 − λ)
+λ
Fγ (z) − Fγ (zt)
z(Fγ (z))′ − tz(Fγ (zt))′
(
)
z(z + γ)
=ϕ
, Fγ (0) = 0 = (Fγ (0))′ − 1
1 + γz
and
[
[
]
]
(1 − t)z(Gγ (z))′
(1 − t)(z 2 (Gγ (z))′′ + z(Gγ (z))′ )
(1 − λ)
+λ
Gγ (z) − Gγ (zt)
z(Gγ (z))′ − tz(Gγ (zt))′
(
)
−z(z + γ)
=ϕ
, Gγ (0) = 0 = (Gγ (0))′ − 1
1 + γz
Obviously, the functions Kδϕ , Fγ , Gγ ∈ M (ϕ, λ, t). Also, we write K ϕ := K2ϕ . If
µ < σ1 or µ > σ2 , then equality holds if and only if f is K ϕ or one of its rotations.
When σ1 < µ < σ2 , then equality holds if and only if f is K3ϕ or one of its rotations.
If µ = σ1 , then equality holds if and only if f is Fγ or one of its rotations. If µ = σ2 ,
then equality holds if and only if f is Gγ or one of its rotations.
If σ1 ≤ µ ≤ σ2 , in view of Lemma 1.2, Theorem 2.1 can be improved.
CERTAIN COEFFICIENT INEQUALITIES FOR SAKAGUCHI TYPE FUNCTIONS...
33
Let f (z) given by (1.1) belongs to M (ϕ, λ, t) and σ3 be given by
[
(
)
]
(1 − t)(1 + λ)2
B2
1 + t (1 + 3λ)
σ3 :=
+ B1
B1 (2 + t)(1 + 2λ) B1
1 − t (1 + λ)2
If σ1 < µ ≤ σ3 , then
|a3 − µa22 |
(
]
[
)
)
(
1
1 + t (1 + 3λ)
1 − t (1 + λ)2
2
2
+ 2 (B1 − B2 )
− B1
+ µB1 |a2 |2
B1
2 + t (1 + 2λ)
2 + t (1 + 2λ)
B1
≤
.
(1 + 2λ)(1 − t)(2 + t)
If σ3 < µ ≤ σ2 , then
|a3 − µa22 |
+
[
(
)
(
)
]
1
1 + t (1 + 3λ)
1 − t (1 + λ)2
2
2
2
+
B
−
µB
(B
+
B
)
1
2
1
1 |a2 |
B12
2 + t (1 + 2λ)
2 + t (1 + 2λ)
B1
≤
.
(1 + 2λ)(1 − t)(2 + t)
Corollary 2.2. For λ = 1 in Theorem 2.1, f (z) given by (1.1) belongs to T (ϕ, t),
then
|a3 − µa22 |

[
(
)
(
)]
3
1
2 1+t
2 2+t

B
+
B
−
µB

2
1 1−t
1 1−t
 3(1−t)(2+t)
4



B1
≤ 3(1−t)(2+t)



[
(
)
(
)]


1
3
2 1+t
2 2+t
−
B
+
B
−
µB
2
1
1
3(1−t)(2+t)
1−t
4
1−t
where
if µ ≤ σ1 ,
if σ1 ≤ µ ≤ σ2 ,
if µ ≥ σ2
[
(
)]
4(1 − t)
B2
1+t
σ1 :=
−1 +
+ B1
3B1 (2 + t)
B1
1−t
and
σ2 :=
[
(
)]
4(1 − t)
B2
1+t
1+
+ B1
3B1 (2 + t)
B1
1−t
The result is sharp.
Also σ3 is given by
σ3 :=
[
(
)]
4(1 − t)
B2
1+t
+ B1
3B1 (2 + t) B1
1−t
If σ1 < µ ≤ σ3 , then
|a3 − µa22 |
(
)
(
)
]
[
1+t
1
4 1−t
24
2
− B1
+ µB1 |a2 |2
+ 2 (B1 − B2 )
B1
3 2+t
3 2+t
B1
≤
.
3(1 − t)(2 + t)
34
B. SRUTHA KEERTHI, V.G. SHANTHI, B. ADOLF STEPHEN
If σ3 < µ ≤ σ2 , then
|a3 − µa22 |
]
[
(
)
(
)
1
4 1−t
1+t
24
2
+ 2 (B1 + B2 )
+ B1
− µB1 |a2 |2
B1
3 2+t
3 2+t
B1
≤
.
3(1 − t)(2 + t)
Remark 2.3. When λ = 0 in Theorem 2.1, f (z) given by (1.1) belongs to S ∗ (ϕ, t)
and the result coincides with a recent result of Goyal et al. [1].
Theorem 2.4. Let ϕ(z) = 1 + B1 z + B2 z 2 + · · · . If f (z) given by (1.1) belongs to
M (ϕ, λ, t), then
B1
|a3 − µa22 | ≤
(1 + 2λ)(1 − t)(2 + t)
}
{ (
)
(
)
B2
1 + t (1 + 3λ)
2 + t (1 + 2λ) max 1, + B1
−
µB
1
B1
1 − t (1 + λ)2
1 − t (1 + λ)2 (2.11)
This result is sharp.
Proof. By applying Lemma 1.3 in (2.10), we obtain the result (2.11).
The result (2.11) is sharp for the function defined by
[
]
[
]
(1 − t)zf ′ (z)
(1 − t)(z 2 f ′′ (z) + zf ′ (z))
(1 − λ)
+λ
= ϕ(z 2 )
f (z) − f (zt)
zf ′ (z) − tzf ′ (zt)
and
[
]
[
]
(1 − t)zf ′ (z)
(1 − t)(z 2 f ′′ (z) + zf ′ (z))
(1 − λ)
+λ
= ϕ(z)
f (z) − f (zt)
zf ′ (z) − tzf ′ (zt)
Corollary 2.5. Let λ = 0 in Theorem 2.4, f (z) given by (1.1) belongs to S ∗ (ϕ, t).
Let ϕ(z) = 1 + B1 z + B2 z 2 + · · · , then
{ (
)
(
)}
B2
B1
1+t
2 + t 2
|a3 − µa2 | ≤
max 1, + B1
− µB1
(1 − t)(2 + t)
B1
1−t
1−t This result is sharp.
Corollary 2.6. Let λ = 1 in Theorem 2.4. Let ϕ(z) = 1 + B1 z + B2 z 2 + · · · and
f (z) given by (1.1) belongs to T (ϕ, t), then
{ (
)
(
)}
B2
B1
1+t
3 2 + t 2
|a3 − µa2 | ≤
max 1, + B1
− µB1
3(1 − t)(2 + t)
B1
1−t
4 1−t This result is sharp.
3. Applications to Functions defined by
Fractional Derivatives
∞
∞
∑
∑
n
For two analytic functions f (z) = z+
an z and g(z) = z+
gn z n , their conn=2
volution
(or
n=2
Hadamard product) is defined to be the function
∞
∑
(f ∗ g)z = z +
an gn z n . For a fixed g ∈ A, let M g (ϕ, λ, t), S g (ϕ, t), T g (ϕ, t)
n=2
CERTAIN COEFFICIENT INEQUALITIES FOR SAKAGUCHI TYPE FUNCTIONS...
35
be the classes of functions f ∈ A for which f ∗ g belongs to M (ϕ, λ, t), S ∗ (ϕ, t) and
T (ϕ, t) respectively.
Definition 3.1. Let f (z) be analytic in a simply connected region of the z-plane
containing the origin. The fractional derivative of f of order γ is defined by
∫ z
d
f (ζ)
1
dζ (0 ≤ γ < 1)
(3.1)
Dzγ f (z) =
Γ(1 − γ) dz 0 (z − ζ)γ
where the multiplicity of (z − ζ)−γ is removed by requiring that log(z − ζ) is real
for (z − ζ) > 0.
Using definition 3.1, Owa and Srivastava (see [4, 5]; see also [9, 10])
introduced a fractional derivative operator Ωγ : A → A defined by
(Ωγ f )(z) = Γ(2 − γ)z γ Dzγ f (z), (γ ̸= 2, 3, 4, . . . ). The classes M γ (ϕ, λ, t), S γ (ϕ, t),
T γ (ϕ, t) consists of functions f ∈ A for which Ωγ f belongs to M (ϕ, λ, t), S ∗ (ϕ, t)
and T (ϕ, t) respectively. The classes M γ (ϕ, λ, t), S γ (ϕ, t), T γ (ϕ, t) is a special case
of
the
classes
M g (ϕ, λ, t),
S g (ϕ, t)
and
T g (ϕ, t)
respectively, when
g(z) = z +
∞
∑
Γ(n + 1)Γ(2 − γ) n
z (z ∈ ∆).
Γ(n + 1 − γ)
n=2
The classes S g (ϕ, t) and S γ (ϕ, t) were studied by Goyal et al. [1].
Now applying Theorem 2.1 for the function (f ∗ g)(z) = z + a2 g2 z 2 + a3 g3 z 3 + · · ·
we get the following theorem after an obvious change of the parameter µ:
Theorem 3.2. Let g(z) = z +
to M g (ϕ, λ, t), then,
∞
∑
gn z n (gn > 0). If f (z) given by (1.1) belongs
n=2
|a3 − µa22 |

[
(
)
(
)
]
1+t (1+3λ)
2+t (1+2λ)
1
2 g3

B2 + B12 1−t

2 − µB1 g 2
2

g
(1+2λ)(1−t)(2+t)
(1+λ)
1−t
(1+λ)
3
2



B1
≤ g3 (1+2λ)(1−t)(2+t)



[
(
)
)
]
(


1
2+t (1+2λ)
2 1+t (1+3λ)
2 g3
−
B
+
B
−
µB
2
2
1 1−t (1+λ)2
1g
g3 (1+2λ)(1−t)(2+t)
1−t (1+λ)2
2
where
η1 :=
and
η2 :=
if µ ≤ η1 ,
if η1 ≤ µ ≤ η2 ,
if µ ≥ η2
[
(
)
]
B2
1 + t (1 + 3λ)
g22 (1 − t)(1 + λ)2
−1 +
+ B1
g3 B1 (2 + t)(1 + 2λ)
B1
1 − t (1 + λ)2
[
(
)
]
B2
1 + t (1 + 3λ)
g22 (1 − t)(1 + λ)2
1+
+ B1
g3 B1 (2 + t)(1 + 2λ)
B1
1 − t (1 + λ)2
The result is sharp.
Since Ωγ f (z) = z +
∞
∑
Γ(n + 1)Γ(2 − γ) n
z ,
Γ(n + 1 − γ)
n=2
we have
g2 :=
2
Γ(3)Γ(2 − γ)
=
Γ(3 − γ)
2−γ
(3.2)
36
B. SRUTHA KEERTHI, V.G. SHANTHI, B. ADOLF STEPHEN
and
g3 :=
Γ(4)Γ(2 − γ)
6
=
Γ(4 − γ)
(2 − γ)(3 − γ)
(3.3)
For g2 , g3 given by (3.2) and (3.3) respectively, Theorem 3.2 reduces to the
following:
Theorem 3.3. Let γ < 2. If f (z) given by (1.1) belongs to M γ (ϕ, λ, t), then
|a3 − µa22 |

[
(
)
(
)
]
(2−γ)(3−γ)
3
2 1+t (1+3λ)
2 (2−γ) 2+t (1+2λ)

B
+
B
−
µB

2
2
2
1
1

6(1+2λ)(1−t)(2+t)
1−t (1+λ)
2
(3−γ) 1−t (1+λ)



B1 (2−γ)(3−γ)
≤ 6(1+2λ)(1−t)(2+t)



[
(
)
(
)
]


3
2 1+t (1+3λ)
2 (2−γ) 2+t (1+2λ)
− (2−γ)(3−γ)
6(1+2λ)(1−t)(2+t) B2 + B1 1−t (1+λ)2 − 2 µB1 (3−γ) 1−t (1+λ)2
where
η1∗
if µ ≤ η1∗ ,
if η1∗ ≤ µ ≤ η2∗ ,
if µ ≥ η2∗ ,
[
(
)
]
2(3 − γ)(1 − t)(1 + λ)2
B2
1 + t (1 + 3λ)
:=
−1 +
+ B1
3(2 − γ)B1 (2 + t)(1 + 2λ)
B1
1 − t (1 + λ)2
and
η2∗
[
(
)
]
2(3 − γ)(1 − t)(1 + λ)2
B2
1 + t (1 + 3λ)
:=
1+
+ B1
3(2 − γ)B1 (2 + t)(1 + 2λ)
B1
1 − t (1 + λ)2
The result is sharp.
Corollary 3.4. For λ = 1 in Theorem 3.2, f (z) given by (1.1) belongs to T g (ϕ, t),
then
|a3 − µa22 |

(
[
)
)]
(
3
1
2+t
2 1+t
2 g3

B
+
B
−
µB

2
2
1
1

3g
(1−t)(2+t)
1−t
4
1−t
g2
3



B1
≤ 3g3 (1−t)(2+t)



[
(
)
(
)]


1
3
2+t
2 1+t
2 g3
−
3g3 (1−t)(2+t) B2 + B1 1−t − 4 µB1 g 2
1−t
2
where
and
if µ ≤ η1 ,
if η1 ≤ µ ≤ η2 ,
if µ ≥ η2 ,
[
(
)]
4g22 (1 − t)
B2
1+t
η1 :=
−1 +
+ B1
3g3 B1 (2 + t)
B1
1−t
[
(
)]
4g22 (1 − t)
B2
1+t
η2 :=
1+
+ B1
3g3 B1 (2 + t)
B1
1−t
The result is sharp.
Remark 3.5. When λ = 0 in Theorem 3.2, f (z) given by (1.1) belongs to S g (ϕ, t)
and the result coincides with a result of Goyal et al. [1].
CERTAIN COEFFICIENT INEQUALITIES FOR SAKAGUCHI TYPE FUNCTIONS...
37
Corollary 3.6. For λ = 1 in Theorem 3.3, f (z) given by (1.1) belongs to T γ (ϕ, t),
then
|a3 − µa22 |

[
(
)
(
)]
(2−γ)(3−γ)
9
2 1+t
2 (2−γ) 2+t

B
+
B
−
µB

2
1 1−t
1 (3−γ) 1−t

18(1−t)(2+t)
8



B1 (2−γ)(3−γ)
≤
18(1−t)(2+t)



[
(
)
(
)]


− (2−γ)(3−γ) B2 + B12 1+t − 9 µB12 (2−γ) 2+t
18(1−t)(2+t)
1−t
8
(3−γ) 1−t
where
η1∗ :=
8
9
(
and
η2∗ :=
8
9
3−γ
2−γ
(
)
3−γ
2−γ
if µ ≤ η1∗ ,
if η1∗ ≤ µ ≤ η2∗ ,
if µ ≥ η2∗ ,
[
(
)]
(1 − t)
B2
1+t
−1 +
+ B1
B1 (2 + t)
B1
1−t
)
[
(
)]
(1 − t)
B2
1+t
1+
+ B1
B1 (2 + t)
B1
1−t
The result is sharp.
Remark 3.7. For λ = 0 in Theorem 3.3, f (z) given by (1.1) belongs to S γ (ϕ, t)
and the result coincides with a result of Goyal et al. [1].
Theorem 3.8. Let g(z) = z +
by ϕ(z) = 1 +
∞
∑
∞
∑
gn z n (gn > 0) and let the function ϕ(z) be given
n=2
Bn z n . If f (z) given by (1.1) belongs to M g (ϕ, λ, t), then
n=1
B1
|a3 − µa22 | ≤
g3 (1 + 2λ)(1 − t)(2 + t)
}
{ (
(
)
)
B2
1 + t (1 + 3λ)
g3 2 + t (1 + 2λ) max 1, + B1
− µB1 2
B1
1 − t (1 + λ)2
g2 1 − t (1 + λ)2 The proof of Theorem 3.8 is similar to Theorem 2.4, so the details are omitted.
Corollary 3.9. Let λ = 0 in Theorem 3.8, f (z) given by (1.1) belongs to S g (ϕ, t),
then
(
{ (
)
)}
B2
B1
1+t
g3 2 + t |a3 − µa22 | ≤
max 1, + B1
− µB1 2
g3 (1 − t)(2 + t)
B1
1−t
g2 1 − t Corollary 3.10. Let λ = 1 in Theorem 3.8, f (z) given by (1.1) belongs to T g (ϕ, t),
then
B1
|a3 − µa22 | ≤
3g3 (1 − t)(2 + t)
(
)
(
)}
{ B2
1+t
3
g3 2 + t + B1
− µB1 2
max 1, B1
1−t
4
g2 1 − t For g2 , g3 given by (3.2) and (3.3) respectively, Theorem 3.8 reduces to the
following:
38
B. SRUTHA KEERTHI, V.G. SHANTHI, B. ADOLF STEPHEN
Theorem 3.11. Let ϕ(z) = 1 + B1 z + B2 z 2 + · · · . If f (z) given by (1.1) belongs
to M γ (ϕ, λ, t), then
B1 (2 − γ)(3 − γ)
|a3 − µa22 | ≤
6(1 + 2λ)(1 − t)(2 + t)
}
(
)
(
)(
)
{ B2
1 + t (1 + 3λ) 3
2−γ
2 + t (1 + 2λ) max 1, + B1
− µB1
B1
1 − t (1 + λ)2
2
3−γ
1 − t (1 + λ)2 The result is sharp.
Corollary 3.12. Let λ = 0 in Theorem 3.11, f (z) given by (1.1) belongs to S γ (ϕ, t),
then
B1 (2 − γ)(3 − γ)
|a3 − µa22 | ≤
6(1 − t)(2 + t)
{ (
)
(
)(
)}
B2
1+t
3
2−γ
2 + t max 1, + B1
.
− µB1
B1
1−t
2
3−γ
1−t Corollary 3.13. Let λ = 1 in Theorem 3.11, f (z) given by (1.1) belongs to
T γ (ϕ, t), then
B1 (2 − γ)(3 − γ)
|a3 − µa22 | ≤
18(1 − t)(2 + t)
{ (
)
(
)(
)}
B2
1+t
9
2−γ
2 + t max 1, + B1
− µB1
.
B1
1−t
8
3−γ
1−t Acknowledgement:
The first author thanks the support provided by Science and Engineering Research
Board , New Delhi - 110 016.Project no:SR/S4/MS:716/10 with titled ”On Classes
of Certain Analytic Univalent Functions and Sakaguchi Type Functions”.
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CERTAIN COEFFICIENT INEQUALITIES FOR SAKAGUCHI TYPE FUNCTIONS...
39
B. Srutha Keerthi
Department of Applied Mathematics, Sri Venkateswara College of Engineering, Sriperumbudur, Chennai - 602 105, India
E-mail address: sruthilaya06@yahoo.co.in
V.G. Shanthi
Department of Mathematics, S.D.N.B. Vaishnav College for Women, Chromepet, Chennai - 600 044, India
E-mail address: vg.shanthi1@gmail.com
B. Adolf Stephen
Department of Mathematics, Madras Christian College, Tambaram, Chennai - 600 059
E-mail address: adolfmcc2003@yahoo.co.in