Directory UMM :Journals:Journal_of_mathematics:AUA:
ACTA UNIVERSITATIS APULENSIS
No 19/2009
CERTAIN COEFFICIENT INEQUALITIES FOR SAKAGUCHI
TYPE FUNCTIONS AND APPLICATIONS TO FRACTIONAL
DERIVATIVE OPERATOR
S. P. Goyal, Pranay Goswami
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-Szeg¨
o inequalities for certain classes of functions defined through fractional
derivatives are obtained.
2000 Mathematics Subject Classification: 30C50; 30C45; 30C80.
Keywords: Sakaguchi functions, Analytic functions, Subordination, Fekete-Szeg¨
o
inequality.
1. Introduction
Let A be the class of analytic functions of the form
f (z) = z +
∞
X
an z n
(z ∈ ∆ := {z ∈ C| |z| < 1})
(1.1)
n=2
and S be the subclass of A consisting of univalentf unctions. 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 analytic function w(z) with w(0) = 0
and |w(z)| < 1 (z ∈ ∆), such that f (z) = g(w(z)), (z ∈ ∆). In particular, if the
function g is univalent in ∆, the above subordination is equivalent to f (0) = g(0)
and f (∆) ⊂ g(∆).
Recently Owa et al. [7,8] introduced and studied a Sakaguchi type class S ∗ (α, t). A
function f (z) ∈ A is said to be in the class S ∗ (α, t) if it satisfies
Re
(1 − t)zf ′ (z)
f (z) − f (zt)
> α,
159
|t| ≤ 1, t 6= 1
(1.2)
S. P. Goyal, P. Goswami - Certain Coefficient Inequalities for Sakaguchi ...
for some α ∈ [0, 1) and for all z ∈ ∆. For α = 0 and t = −1, we get the class
the class S ∗ (0, −1) studied be Sakaguchi [9]. A function f (z) ∈ S ∗ (α, −1) is called
Sakaguchi function of order α.
In this paper, we define the following class S ∗ (φ, t), which is generalization of
the class S ∗ (α, 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. Then function f ∈ A is
in the class S ∗ (φ, t) if
(1 − t)zf ′ (z)
≺ φ(z),
f (z) − f (zt)
|t| ≤ 1, t 6= 1
(1.5)
Again T (φ, t) denote the subclass of A consisting functions f (z) such that zf ′ (z) ∈
S ∗ (φ, t).
When φ(z) = (1 + Az)/(1 + Bz), (−1 ≤ B < A ≤ 1), we denote the subclasses
S ∗ (φ, t) and T (φ, t) by S ∗ [A, B, t] and T [A, B, t] respectively.
Obviously S ∗ (φ, 0) ≡ S ∗ (φ). When t = −1, then S ∗ (φ, −1) ≡ Ss∗ (φ), which is a
known class studied by Shanmugam et al. [10]. For t = 0 and φ(z) = (1 + Az)/(1 +
Bz),
(−1 ≤ B < A ≤ 1), the subclass S ∗ (φ, t) reduces to the class S ∗ [A, B]
studied by Janowski [3]. For 0 ≤ α < 1 let S ∗ (α, t) := S ∗ [1 − 2α, −1; t], which is a
, our
known class studied by Owa et al. [8]. Also, for t = −1 and φ(z) = 1+(1−2α)z
1−z
class reduces to a known class S(α, −1) studied by Cho et al. ([1], see also [8]).
In the present paper, we obtain the Fekete-Szeg¨
o inequality for the functions in
∗
the subclass S (φ, t). We also give application of our results to certain functions
defined through convolution (or Hadamard product) and in particular, we consider
the class S λ (φ, t) defined by fractional derivatives.
To prove our main results, we need the following lemma:
Lemma 1.2. ([4]) If p(z) = 1 + c1 z + c2 z 2 ... is an analytic function with positive
real part in ∆, then
|c2 − vc21 | ≤
−4v + 2
2
4v − 2
if v ≤ 0,
if 0 ≤ v ≤ 1,
if v ≥ 1.
When v < 0 or v > 1, the equality holds if and only if p(z) is (1 + z)/(1 − z) or
one of its rotations. If 0 < v < 1, then the equality holds if and only if p(z) is
(1 + z 2 )/(1 − z 2 ) or one of its rotations. If v = 0, the equality holds if and only if
p(z) =
1+z
1 1
+ λ
+
2 2
1−z
1 1
1−z
− λ
2 2
1+z
160
(0 ≤ λ ≤ 1)
S. P. Goyal, P. Goswami - Certain Coefficient Inequalities for Sakaguchi ...
or one of its rotations. If v = 1, the equality holds if and only if p(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:
(0 < v ≤ 1/2)
|c2 − vc21 | + v|c1 |2 ≤ 2
and
|c2 − vc21 | + (1 − v)|c1 |2 ≤ 2
(1/2 < v ≤ 1).
2. Main Results
Our main result is contained in the following Theorem :
Theorem 2.1. If f (z) given by (1.1) belongs to S ∗ (φ, t), then
|a3 −
where
µa22 |
≤
h
i
1
2 1+t
2 2+t
if
(t+2)(1−t) B2 + B1 1−t − µB1 1−t
B1
if σ1
(t+2)(1−t) h
i
1+t
1
B2 + B12 1−t
− µB12 2+t
if
− (t+2)(1−t)
1−t
σ1 :=
and
B2
1+t
(1 − t)
+ B1
−1 +
B1 (2 + t)
B1
1−t
(1 − t)
B2
1+t
1+
+ B1
σ2 :=
B1 (2 + t)
B1
1−t
µ ≤ σ1 ,
≤ µ ≤ σ2 ,
µ ≥ σ2 ,
,
.
The result is sharp.
Proof. Let f ∈ S ∗ (φ, t). Then there exists a Schwarz function w(z) ∈ A such that
(1 − t)zf ′ (z)
= φ(w(z))
f (z) − f (zt)
(z ∈ ∆;
|t| ≤ 1, t 6= 1)
(2.1)
If p1 (z) is analytic and has positive real part in ∆ and p1 (0) = 1, then
1 + w(z)
= 1 + c1 z + c2 z 2 + .....
1 − w(z)
p1 (z) =
(z ∈ ∆).
(2.2)
From (2.2), we obtain
c2
c1
1
c2 − 1
w(z) = z +
2
2
2
Let
p(z) =
!
z 2 + ...
(1 − t)zf ′ (z)
= 1 + b1 z + b2 z 2 + ...
f (z) − f (tz)
161
(2.3)
(z ∈ ∆),
(2.4)
S. P. Goyal, P. Goswami - Certain Coefficient Inequalities for Sakaguchi ...
which gives
b1 = (1 − t)a2 and b2 = (t2 − 1)a22 + (2 − t − t2 )a3 .
(2.5)
Since φ(z) is univalent and p ≺ φ, therefore using (2.3), we obtain
B1 c1
p(z) = φ(w(z)) = 1 +
z+
2
(
c2
1
c2 − 1
2
2
!
)
1
B1 + c21 B2 z 2 + ... (z ∈ ∆), (2.6)
4
Now from (2.4), (2.5) and (2.6), we have
B1 c1
(1−t)a2 =
,
2
c2
1
c2 − 1
2
2
!
1
B1 + c21 B2 = (t2 −1)a22 +(2−t−t2 )a3 , |t| ≤ 1, t 6= 1.
4
Therefore we have
a3 − µa22 =
where
h
i
B1
c2 − vc21 ,
2(1 − t)(t + 2)
(2.7)
B2
1+t
2+t
1
v :=
1−
− B1
+ µB1
2
B1
1−t
1−t
Our result now follows by an application of Lemma 1.2. To shows that these bounds
are sharp, we define the functions Kφn (n = 2, 3...) by
′
(1 − t)zKφn (z)
′
Kφn (z) − Kφn (tz)
= φ(z n−1 ),
Kφn (0) = 0 = [Kφn ]′ (0) − 1
and the function F λ and Gλ (0 ≤ λ ≤ 1) by
′
z(z + λ)
(1 − t)zFλ (z)
=φ
,
Fλ (z) − Fλ (tz)
1 + λz
and
′
−z(z + λ)
(1 − t)zGλ (z)
=φ
,
Gλ (z) − Gλ (tz)
1 + λz
Fλ (0) = 0 = [Fλ ]′ (0) − 1
Gλ (0) = 0 = [Gλ ]′ (0) − 1.
Obviously the functions Kφn , Fλ , Gλ ∈ S ∗ (φ, t). Also we write Kφ := Kφ2 . If µ < σ1
or µ > σ2 , then equality holds if and only if f is Kφ or one of its rotations. When
σ1 < µ < σ2 , the equality hods if and only if f is Kφ3 or one of its rotations. µ = σ1
then equality holds if and only if f is Fλ or one of its rotations. µ = σ2 then equality
holds if and only if f is Gλ or one of its rotations.
If σ1 ≤ µ ≤ σ2 , in view of Lemma 1.2, Theorem 2.1 can be improved.
Theorem 2.2. Let f (z) given by (1.1) belongs to S ∗ (φ, t). and σ3 be given by
1
σ3 :=
B1
1−t
2+t
B2
1+t
− B1
B1
1−t
162
.
S. P. Goyal, P. Goswami - Certain Coefficient Inequalities for Sakaguchi ...
If σ1 < µ ≤ σ3 , then
1
1−t
2 1+t
2
2
− B1
+ µB1 |a2 |2 ≤
a3 − µa2 + 2 (B1 − B2 )
B1
.
(1 − t)(t + 2)
1
1−t
2
2 1+t
2
+ B1
+ µB1 |a2 |2 ≤
a3 − µa2 + 2 (B1 + B2 )
B1
(1 − t)(t + 2)
2+t
B1
2+t
If σ3 < µ ≤ σ2 , then
2+t
B1
2+t
Example 2.3. Let −1 ≤ B < A ≤ 1. If f (z) ∈ A belongs to S ∗ [A, B; t], then
|a3 − µa22 | ≤
where
B−A
(2+t)(1−t)
A−B
(2+t)(1−t)
A−B
(t+2)(1−t)
σ1∗
h
h
B − (A − B)
1+t
1−t
B − (A − B)
1+t
1−t
+ µ(A − B)
2+t
1−t
+ µ(A − B)
2+t
1−t
(1 − t)
1+B
=
−
+
(2 + t)
(A − B)
and
σ2∗ =
(1 − t) 1 − B
+
(2 + t) (A − B)
1+t
1−t
1+t
1−t
i
if µ ≤ σ1∗ ,
σ1∗ ≤ µ ≤ σ2∗ ,
i
if
if
µ ≤ σ2∗ ,
,
.
In particular, if f ∈ S(α, t), then
|a3 − µa22 | ≤
where
h
i
2(1−α)
2+t
1+t
if µ ≤ σ1∗∗ ,
(2+t)(1−t) 1 + 2(1 − α) 1−t − 2µ(1 − α) 1−t
2(1−α)
if σ1∗∗ ≤ µ ≤ σ2∗∗ ,
(2+t)(1−t) h
i
2+t
1−α
−
µα
if µ ≥ σ2∗∗ ,
(1 − 2α) − α 1+t
2 (t+2)(1−t)
1−t
1−t
σ1∗∗
and
σ2∗∗
1+t
=−
2+t
(1 − t)
1
=
−
(2 + t) (1 − α)
The results are sharp.
163
1+t
1−t
S. P. Goyal, P. Goswami - Certain Coefficient Inequalities for Sakaguchi ...
3. Applications to functions defined by fractional derivatives
For two analytic functions f (z) = z +
∞
P
an z n and g(z) = z +
n=0
∞
P
gn z n , their
n=0
convolution (or Hadamard product) is defined to be the function (f ∗ g)(z) = z +
∞
P
an gn z n . For a fixed g ∈ A, let S g (φ, t) be the class of functions f ∈ A for which
n=0
(f ∗ g) ∈ S ∗ (φ, t).
Definition 3.1. Let f (z) be analytic in a simply connected region of the z-plane
containing origin. The fractional derivative of f of order λ is defined by
λ
0 Dz f (z)
1
d
:=
Γ(1 − λ) dz
Zz
(z − ζ)−λ f (ζ)dζ
(0 ≤ λ < 1),
(3.1)
0
where the multiplicity of (z − ζ)−λ is removed by requiring that log(z − ζ) is real for
(z − ζ) > 0
Using definition 3.1, Owa and Srivastava (see [5],[6]; see also [11], [12]) introduced
a fractional derivative operator Ωλ : A −→ A defined by
(Ωλ f )(z) = Γ(2 − λ)z λ 0 Dzλ f (z),
(λ 6= 2, 3, 4, ...)
The class S λ (φ, t) consists of the functions f ∈ A for which Ωλ f ∈ S ∗ (φ, t). The
class S λ (φ, t) is a special case of the class S g (φ, t) when
g(z) = z +
∞
X
Γ(n + 1)Γ(2 − λ)
Γ(n + 1 − λ)
n=2
zn
(z ∈ ∆).
Now applying Theorem 2.1 for the function (f ∗ g)(z) = z + g2 a2 z 2 + g3 a3 z 3 + ...,
we get following theorem after an obvious change of the parameter µ:
Theorem 3.2 Let g(z) = z +
|a3 −
where
gn z n (gn > 0). If f (z) is given by (1.1) belongs to
n=2
S g (φ, t), then
µa22 |
∞
P
≤
h
i
g3 2 2+t
1
2 1+t
g3 (2+t)(1−t) B2 + B1 1−t − µ g22 B1 1−t
B1
if
g3 (2+t)(1−t) h
i
g3 2 2+t
1+t
1
2
− g3 (2+t)(1−t) B2 + B1 1−t − µ g2 B1 1−t
2
η1 :=
η2 :=
g22
g3 B1
g22
g3 B1
h
i
1+t
1−t
2
−1 + B
1 1−t
B1 + B
2+t h
i
B2
1−t
1+t
+
B
1
+
1
2+t
B1
1−t
164
if µ ≤ η1 ,
η1 ≤ µ ≤ η2 ,
if µ ≥ η2 ,
S. P. Goyal, P. Goswami - Certain Coefficient Inequalities for Sakaguchi ...
The result is sharp.
Since
Ωλ f (z) = z +
∞
X
Γ(n + 1)Γ(2 − λ)
Γ(n + 1 − λ)
n=2
We have
2
Γ(3)Γ(2 − λ)
=
Γ(3 − λ)
2−λ
(3.2)
Γ(4)Γ(2 − λ)
6
=
.
Γ(4 − λ)
(2 − λ)(3 − λ)
(3.3)
g2 :=
and
g3 :=
zn,
For g2 , g3 given by (3.2) and (3.3) respectively, Theorem 3.2 reduces to the following
:
Theorem 3.3. Let λ < 2. If f (z) given by (1.1) belongs to S λ (φ, t). Then
|a3 − µa22 | ≤
where
h
i
(2−λ)(3−λ)
3
3−λ
2 2+t
2 1+t
if µ ≤ η1∗ ,
6(2+t)(1−t) B2 + B1 1−t − 2 µ 2−λ B1 1−t
(2−λ)(3−λ)
if η1∗ ≤ µ ≤ η2∗ ,
6(2+t)(1−t) B1h
i
(2−λ)(3−λ)
3−λ
3
2 2+t
− 6(2+t)(1−t)
µ
B2 + B12 1+t
−
B
if µ ≥ η1∗ ,
1 1−t
1−t
2
2−λ
η1 :=
η2 :=
(3−λ)
(2−λ)B1
(3−λ)
(2−λ)B1
h
i
B2
1−t
+ B1 1+t
−1 + B
1
2+t h
1−t
i
B2
1−t
1+t
+
B
1
+
.
1 1−t
2+t
B1
The result is sharp.
Remark. For t = −1 in aforementioned Theorems 2.1, 2.2, 3.2, 3.3 and example
2.3, we arrive at the results obtained recently by Shanmugham et al. [10].
References
[1] N. E. Cho, O. S. Kwon and S. Owa, Certain subclasses of Sakaguchi functions,
SEA Bull. Math.,17 (1993), 121-126.
[2] A. W. Goodman, Uniformaly convex functions, Ann. Polon. Math., 56
(1991), 87-92.
[3] W. Janowski, Some extremal problems for certain families of analytic functions, Bull. Acad. Plolon. Sci. Ser. Sci. Math. Astronomy , 21 (1973), 17-25.
[4] W. Ma and D. Minda, A unified treatment of some special classes of univalent
functions, in : Proceedings of Conference of Complex Analysis, Z. Li., F. Ren, L
Yang, and S. Zhang (Eds.), Intenational Press (1994), 157-169.
[5] S. Owa, On distortion theorems I, Kyunpook Math. J., 18 (1) (1978), 53-59.
165
S. P. Goyal, P. Goswami - Certain Coefficient Inequalities for Sakaguchi ...
[6] S. Owa and H. M. Srivastava, Univalent and starlike generalized hypegeometric
functions, Canad. J. Math., 39 (5) (1987), 1057-1077.
[7] S. Owa, T. Sekine, and R. Yamakawa, Notes on Sakaguchi type functions,
RIMS Kokyuroku, 1414 (2005), 76 - 82.
[8] S. Owa, T. Sekine, and R. Yamakawa, On Sakaguchi type functions, Appl.
Math. Comput., 187 (2007), 356-361.
[9] K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan, 11
(1959), 72-75.
[10] T. N. Shanmugham, C. Ramachandran, and V. Ravichandran, Fekete - Szeg¨
o
problem for a subclasses of starlike functions with respect to symmetric points, Bull.
Korean Math. Soc. 43 (3) (2006), 589-598.
[11] H. M. Srivastava and S. Owa, An application of fractional derivative, Math.
Japon., 29 (3) (1984), 383-389.
[12] H. M. Srivastava and S. Owa, Univalent Functions, Fractional Calculus and
Their Applications, Halsted Press/ John Wiley and Sons, Chichester / New York,
1989.
S. P. Goyal
Department of Mathematics
University of Rajasthan
Jaipur (INDIA) - 302055
email:[email protected]
Pranay Goswami
Department of Mathematics
Amity University Rajasthan
Jaipur (INDIA) - 302002
email:[email protected]
166
No 19/2009
CERTAIN COEFFICIENT INEQUALITIES FOR SAKAGUCHI
TYPE FUNCTIONS AND APPLICATIONS TO FRACTIONAL
DERIVATIVE OPERATOR
S. P. Goyal, Pranay Goswami
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-Szeg¨
o inequalities for certain classes of functions defined through fractional
derivatives are obtained.
2000 Mathematics Subject Classification: 30C50; 30C45; 30C80.
Keywords: Sakaguchi functions, Analytic functions, Subordination, Fekete-Szeg¨
o
inequality.
1. Introduction
Let A be the class of analytic functions of the form
f (z) = z +
∞
X
an z n
(z ∈ ∆ := {z ∈ C| |z| < 1})
(1.1)
n=2
and S be the subclass of A consisting of univalentf unctions. 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 analytic function w(z) with w(0) = 0
and |w(z)| < 1 (z ∈ ∆), such that f (z) = g(w(z)), (z ∈ ∆). In particular, if the
function g is univalent in ∆, the above subordination is equivalent to f (0) = g(0)
and f (∆) ⊂ g(∆).
Recently Owa et al. [7,8] introduced and studied a Sakaguchi type class S ∗ (α, t). A
function f (z) ∈ A is said to be in the class S ∗ (α, t) if it satisfies
Re
(1 − t)zf ′ (z)
f (z) − f (zt)
> α,
159
|t| ≤ 1, t 6= 1
(1.2)
S. P. Goyal, P. Goswami - Certain Coefficient Inequalities for Sakaguchi ...
for some α ∈ [0, 1) and for all z ∈ ∆. For α = 0 and t = −1, we get the class
the class S ∗ (0, −1) studied be Sakaguchi [9]. A function f (z) ∈ S ∗ (α, −1) is called
Sakaguchi function of order α.
In this paper, we define the following class S ∗ (φ, t), which is generalization of
the class S ∗ (α, 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. Then function f ∈ A is
in the class S ∗ (φ, t) if
(1 − t)zf ′ (z)
≺ φ(z),
f (z) − f (zt)
|t| ≤ 1, t 6= 1
(1.5)
Again T (φ, t) denote the subclass of A consisting functions f (z) such that zf ′ (z) ∈
S ∗ (φ, t).
When φ(z) = (1 + Az)/(1 + Bz), (−1 ≤ B < A ≤ 1), we denote the subclasses
S ∗ (φ, t) and T (φ, t) by S ∗ [A, B, t] and T [A, B, t] respectively.
Obviously S ∗ (φ, 0) ≡ S ∗ (φ). When t = −1, then S ∗ (φ, −1) ≡ Ss∗ (φ), which is a
known class studied by Shanmugam et al. [10]. For t = 0 and φ(z) = (1 + Az)/(1 +
Bz),
(−1 ≤ B < A ≤ 1), the subclass S ∗ (φ, t) reduces to the class S ∗ [A, B]
studied by Janowski [3]. For 0 ≤ α < 1 let S ∗ (α, t) := S ∗ [1 − 2α, −1; t], which is a
, our
known class studied by Owa et al. [8]. Also, for t = −1 and φ(z) = 1+(1−2α)z
1−z
class reduces to a known class S(α, −1) studied by Cho et al. ([1], see also [8]).
In the present paper, we obtain the Fekete-Szeg¨
o inequality for the functions in
∗
the subclass S (φ, t). We also give application of our results to certain functions
defined through convolution (or Hadamard product) and in particular, we consider
the class S λ (φ, t) defined by fractional derivatives.
To prove our main results, we need the following lemma:
Lemma 1.2. ([4]) If p(z) = 1 + c1 z + c2 z 2 ... is an analytic function with positive
real part in ∆, then
|c2 − vc21 | ≤
−4v + 2
2
4v − 2
if v ≤ 0,
if 0 ≤ v ≤ 1,
if v ≥ 1.
When v < 0 or v > 1, the equality holds if and only if p(z) is (1 + z)/(1 − z) or
one of its rotations. If 0 < v < 1, then the equality holds if and only if p(z) is
(1 + z 2 )/(1 − z 2 ) or one of its rotations. If v = 0, the equality holds if and only if
p(z) =
1+z
1 1
+ λ
+
2 2
1−z
1 1
1−z
− λ
2 2
1+z
160
(0 ≤ λ ≤ 1)
S. P. Goyal, P. Goswami - Certain Coefficient Inequalities for Sakaguchi ...
or one of its rotations. If v = 1, the equality holds if and only if p(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:
(0 < v ≤ 1/2)
|c2 − vc21 | + v|c1 |2 ≤ 2
and
|c2 − vc21 | + (1 − v)|c1 |2 ≤ 2
(1/2 < v ≤ 1).
2. Main Results
Our main result is contained in the following Theorem :
Theorem 2.1. If f (z) given by (1.1) belongs to S ∗ (φ, t), then
|a3 −
where
µa22 |
≤
h
i
1
2 1+t
2 2+t
if
(t+2)(1−t) B2 + B1 1−t − µB1 1−t
B1
if σ1
(t+2)(1−t) h
i
1+t
1
B2 + B12 1−t
− µB12 2+t
if
− (t+2)(1−t)
1−t
σ1 :=
and
B2
1+t
(1 − t)
+ B1
−1 +
B1 (2 + t)
B1
1−t
(1 − t)
B2
1+t
1+
+ B1
σ2 :=
B1 (2 + t)
B1
1−t
µ ≤ σ1 ,
≤ µ ≤ σ2 ,
µ ≥ σ2 ,
,
.
The result is sharp.
Proof. Let f ∈ S ∗ (φ, t). Then there exists a Schwarz function w(z) ∈ A such that
(1 − t)zf ′ (z)
= φ(w(z))
f (z) − f (zt)
(z ∈ ∆;
|t| ≤ 1, t 6= 1)
(2.1)
If p1 (z) is analytic and has positive real part in ∆ and p1 (0) = 1, then
1 + w(z)
= 1 + c1 z + c2 z 2 + .....
1 − w(z)
p1 (z) =
(z ∈ ∆).
(2.2)
From (2.2), we obtain
c2
c1
1
c2 − 1
w(z) = z +
2
2
2
Let
p(z) =
!
z 2 + ...
(1 − t)zf ′ (z)
= 1 + b1 z + b2 z 2 + ...
f (z) − f (tz)
161
(2.3)
(z ∈ ∆),
(2.4)
S. P. Goyal, P. Goswami - Certain Coefficient Inequalities for Sakaguchi ...
which gives
b1 = (1 − t)a2 and b2 = (t2 − 1)a22 + (2 − t − t2 )a3 .
(2.5)
Since φ(z) is univalent and p ≺ φ, therefore using (2.3), we obtain
B1 c1
p(z) = φ(w(z)) = 1 +
z+
2
(
c2
1
c2 − 1
2
2
!
)
1
B1 + c21 B2 z 2 + ... (z ∈ ∆), (2.6)
4
Now from (2.4), (2.5) and (2.6), we have
B1 c1
(1−t)a2 =
,
2
c2
1
c2 − 1
2
2
!
1
B1 + c21 B2 = (t2 −1)a22 +(2−t−t2 )a3 , |t| ≤ 1, t 6= 1.
4
Therefore we have
a3 − µa22 =
where
h
i
B1
c2 − vc21 ,
2(1 − t)(t + 2)
(2.7)
B2
1+t
2+t
1
v :=
1−
− B1
+ µB1
2
B1
1−t
1−t
Our result now follows by an application of Lemma 1.2. To shows that these bounds
are sharp, we define the functions Kφn (n = 2, 3...) by
′
(1 − t)zKφn (z)
′
Kφn (z) − Kφn (tz)
= φ(z n−1 ),
Kφn (0) = 0 = [Kφn ]′ (0) − 1
and the function F λ and Gλ (0 ≤ λ ≤ 1) by
′
z(z + λ)
(1 − t)zFλ (z)
=φ
,
Fλ (z) − Fλ (tz)
1 + λz
and
′
−z(z + λ)
(1 − t)zGλ (z)
=φ
,
Gλ (z) − Gλ (tz)
1 + λz
Fλ (0) = 0 = [Fλ ]′ (0) − 1
Gλ (0) = 0 = [Gλ ]′ (0) − 1.
Obviously the functions Kφn , Fλ , Gλ ∈ S ∗ (φ, t). Also we write Kφ := Kφ2 . If µ < σ1
or µ > σ2 , then equality holds if and only if f is Kφ or one of its rotations. When
σ1 < µ < σ2 , the equality hods if and only if f is Kφ3 or one of its rotations. µ = σ1
then equality holds if and only if f is Fλ or one of its rotations. µ = σ2 then equality
holds if and only if f is Gλ or one of its rotations.
If σ1 ≤ µ ≤ σ2 , in view of Lemma 1.2, Theorem 2.1 can be improved.
Theorem 2.2. Let f (z) given by (1.1) belongs to S ∗ (φ, t). and σ3 be given by
1
σ3 :=
B1
1−t
2+t
B2
1+t
− B1
B1
1−t
162
.
S. P. Goyal, P. Goswami - Certain Coefficient Inequalities for Sakaguchi ...
If σ1 < µ ≤ σ3 , then
1
1−t
2 1+t
2
2
− B1
+ µB1 |a2 |2 ≤
a3 − µa2 + 2 (B1 − B2 )
B1
.
(1 − t)(t + 2)
1
1−t
2
2 1+t
2
+ B1
+ µB1 |a2 |2 ≤
a3 − µa2 + 2 (B1 + B2 )
B1
(1 − t)(t + 2)
2+t
B1
2+t
If σ3 < µ ≤ σ2 , then
2+t
B1
2+t
Example 2.3. Let −1 ≤ B < A ≤ 1. If f (z) ∈ A belongs to S ∗ [A, B; t], then
|a3 − µa22 | ≤
where
B−A
(2+t)(1−t)
A−B
(2+t)(1−t)
A−B
(t+2)(1−t)
σ1∗
h
h
B − (A − B)
1+t
1−t
B − (A − B)
1+t
1−t
+ µ(A − B)
2+t
1−t
+ µ(A − B)
2+t
1−t
(1 − t)
1+B
=
−
+
(2 + t)
(A − B)
and
σ2∗ =
(1 − t) 1 − B
+
(2 + t) (A − B)
1+t
1−t
1+t
1−t
i
if µ ≤ σ1∗ ,
σ1∗ ≤ µ ≤ σ2∗ ,
i
if
if
µ ≤ σ2∗ ,
,
.
In particular, if f ∈ S(α, t), then
|a3 − µa22 | ≤
where
h
i
2(1−α)
2+t
1+t
if µ ≤ σ1∗∗ ,
(2+t)(1−t) 1 + 2(1 − α) 1−t − 2µ(1 − α) 1−t
2(1−α)
if σ1∗∗ ≤ µ ≤ σ2∗∗ ,
(2+t)(1−t) h
i
2+t
1−α
−
µα
if µ ≥ σ2∗∗ ,
(1 − 2α) − α 1+t
2 (t+2)(1−t)
1−t
1−t
σ1∗∗
and
σ2∗∗
1+t
=−
2+t
(1 − t)
1
=
−
(2 + t) (1 − α)
The results are sharp.
163
1+t
1−t
S. P. Goyal, P. Goswami - Certain Coefficient Inequalities for Sakaguchi ...
3. Applications to functions defined by fractional derivatives
For two analytic functions f (z) = z +
∞
P
an z n and g(z) = z +
n=0
∞
P
gn z n , their
n=0
convolution (or Hadamard product) is defined to be the function (f ∗ g)(z) = z +
∞
P
an gn z n . For a fixed g ∈ A, let S g (φ, t) be the class of functions f ∈ A for which
n=0
(f ∗ g) ∈ S ∗ (φ, t).
Definition 3.1. Let f (z) be analytic in a simply connected region of the z-plane
containing origin. The fractional derivative of f of order λ is defined by
λ
0 Dz f (z)
1
d
:=
Γ(1 − λ) dz
Zz
(z − ζ)−λ f (ζ)dζ
(0 ≤ λ < 1),
(3.1)
0
where the multiplicity of (z − ζ)−λ is removed by requiring that log(z − ζ) is real for
(z − ζ) > 0
Using definition 3.1, Owa and Srivastava (see [5],[6]; see also [11], [12]) introduced
a fractional derivative operator Ωλ : A −→ A defined by
(Ωλ f )(z) = Γ(2 − λ)z λ 0 Dzλ f (z),
(λ 6= 2, 3, 4, ...)
The class S λ (φ, t) consists of the functions f ∈ A for which Ωλ f ∈ S ∗ (φ, t). The
class S λ (φ, t) is a special case of the class S g (φ, t) when
g(z) = z +
∞
X
Γ(n + 1)Γ(2 − λ)
Γ(n + 1 − λ)
n=2
zn
(z ∈ ∆).
Now applying Theorem 2.1 for the function (f ∗ g)(z) = z + g2 a2 z 2 + g3 a3 z 3 + ...,
we get following theorem after an obvious change of the parameter µ:
Theorem 3.2 Let g(z) = z +
|a3 −
where
gn z n (gn > 0). If f (z) is given by (1.1) belongs to
n=2
S g (φ, t), then
µa22 |
∞
P
≤
h
i
g3 2 2+t
1
2 1+t
g3 (2+t)(1−t) B2 + B1 1−t − µ g22 B1 1−t
B1
if
g3 (2+t)(1−t) h
i
g3 2 2+t
1+t
1
2
− g3 (2+t)(1−t) B2 + B1 1−t − µ g2 B1 1−t
2
η1 :=
η2 :=
g22
g3 B1
g22
g3 B1
h
i
1+t
1−t
2
−1 + B
1 1−t
B1 + B
2+t h
i
B2
1−t
1+t
+
B
1
+
1
2+t
B1
1−t
164
if µ ≤ η1 ,
η1 ≤ µ ≤ η2 ,
if µ ≥ η2 ,
S. P. Goyal, P. Goswami - Certain Coefficient Inequalities for Sakaguchi ...
The result is sharp.
Since
Ωλ f (z) = z +
∞
X
Γ(n + 1)Γ(2 − λ)
Γ(n + 1 − λ)
n=2
We have
2
Γ(3)Γ(2 − λ)
=
Γ(3 − λ)
2−λ
(3.2)
Γ(4)Γ(2 − λ)
6
=
.
Γ(4 − λ)
(2 − λ)(3 − λ)
(3.3)
g2 :=
and
g3 :=
zn,
For g2 , g3 given by (3.2) and (3.3) respectively, Theorem 3.2 reduces to the following
:
Theorem 3.3. Let λ < 2. If f (z) given by (1.1) belongs to S λ (φ, t). Then
|a3 − µa22 | ≤
where
h
i
(2−λ)(3−λ)
3
3−λ
2 2+t
2 1+t
if µ ≤ η1∗ ,
6(2+t)(1−t) B2 + B1 1−t − 2 µ 2−λ B1 1−t
(2−λ)(3−λ)
if η1∗ ≤ µ ≤ η2∗ ,
6(2+t)(1−t) B1h
i
(2−λ)(3−λ)
3−λ
3
2 2+t
− 6(2+t)(1−t)
µ
B2 + B12 1+t
−
B
if µ ≥ η1∗ ,
1 1−t
1−t
2
2−λ
η1 :=
η2 :=
(3−λ)
(2−λ)B1
(3−λ)
(2−λ)B1
h
i
B2
1−t
+ B1 1+t
−1 + B
1
2+t h
1−t
i
B2
1−t
1+t
+
B
1
+
.
1 1−t
2+t
B1
The result is sharp.
Remark. For t = −1 in aforementioned Theorems 2.1, 2.2, 3.2, 3.3 and example
2.3, we arrive at the results obtained recently by Shanmugham et al. [10].
References
[1] N. E. Cho, O. S. Kwon and S. Owa, Certain subclasses of Sakaguchi functions,
SEA Bull. Math.,17 (1993), 121-126.
[2] A. W. Goodman, Uniformaly convex functions, Ann. Polon. Math., 56
(1991), 87-92.
[3] W. Janowski, Some extremal problems for certain families of analytic functions, Bull. Acad. Plolon. Sci. Ser. Sci. Math. Astronomy , 21 (1973), 17-25.
[4] W. Ma and D. Minda, A unified treatment of some special classes of univalent
functions, in : Proceedings of Conference of Complex Analysis, Z. Li., F. Ren, L
Yang, and S. Zhang (Eds.), Intenational Press (1994), 157-169.
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165
S. P. Goyal, P. Goswami - Certain Coefficient Inequalities for Sakaguchi ...
[6] S. Owa and H. M. Srivastava, Univalent and starlike generalized hypegeometric
functions, Canad. J. Math., 39 (5) (1987), 1057-1077.
[7] S. Owa, T. Sekine, and R. Yamakawa, Notes on Sakaguchi type functions,
RIMS Kokyuroku, 1414 (2005), 76 - 82.
[8] S. Owa, T. Sekine, and R. Yamakawa, On Sakaguchi type functions, Appl.
Math. Comput., 187 (2007), 356-361.
[9] K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan, 11
(1959), 72-75.
[10] T. N. Shanmugham, C. Ramachandran, and V. Ravichandran, Fekete - Szeg¨
o
problem for a subclasses of starlike functions with respect to symmetric points, Bull.
Korean Math. Soc. 43 (3) (2006), 589-598.
[11] H. M. Srivastava and S. Owa, An application of fractional derivative, Math.
Japon., 29 (3) (1984), 383-389.
[12] H. M. Srivastava and S. Owa, Univalent Functions, Fractional Calculus and
Their Applications, Halsted Press/ John Wiley and Sons, Chichester / New York,
1989.
S. P. Goyal
Department of Mathematics
University of Rajasthan
Jaipur (INDIA) - 302055
email:[email protected]
Pranay Goswami
Department of Mathematics
Amity University Rajasthan
Jaipur (INDIA) - 302002
email:[email protected]
166