Paper-6-20-2009. 108KB Jun 04 2011 12:03:14 AM
ACTA UNIVERSITATIS APULENSIS
No 20/2009
SUBORDINATION RESULTS FOR CERTAIN CLASSES OF
ANALYTIC FUNCTIONS ASSOCIATED WITH THE
CONVOLUTION STRUCTURE
V. G. Gupta and Bhavna Shrama
Abstract. The main object of the present paper is to derive subordination
property of the class which was very recently PTg (λ, α, β, γ) introduced by Murugusundaramoorthy and Joshi (J. Math. Ineq., 3(1)(2009), pp. 293-303).
2000 Mathematics Subject Classification: 30C45.
Keywords: Univalent functions, Convolution, Subordination .
1. Introduction and Preliminaries
Let A be the class of analytic functions of the form
f (z) = z +
∞
X
ak z k
(z ∈ ∆ := {z ∈ C| |z| < 1})
(1.1)
k=2
which are analytic and univalent in the open unit disk. If U = {z ∈ C : |z| < 1} is
given by (1.1) and g ∈ A is given by
g(z) = z +
∞
X
bk z k
(1.2)
k=2
the Hadamard product (or convolution ) (f ∗ g) of f and g is given by
(f ∗ g)(z) = z +
∞
X
k=2
49
ak bk z k , z ∈ U
(1.3)
V. G. Gupta, B. Shrama - Subordination results for certain classes of ...
Very recently, by using Hadamard product (or convolution) Murugusundaramoorthy
and Joshi [5], introduced the subclass Pg (λ, α, β, γ) of A consisting the functions of
the form (1.1) and satisfying the following inequality,
Jg,λ (z) − 1
< β, λ ≥ 0, 0 ≤ α < 1, 0 0
(z ∈ U)
(1.8)
k=1
To prove our main result, we shall required the following lemma due to Murusundaramoorthy and Joshi.
Lemma 1.2 (see [5]). Let the function f be defined by (1.4) then PTg (λ, α, β, γ) if
and only if
∞
X
(1 + λ(k − 1))[1 + β(2γ − 1)]ak bk ≤ 2βγ(1 − α)
(1.9)
k=2
2. Subordination theorem
Theorem 2.1.Let the function f ∈ PTg (λ, α, β, γ) satisfy the inequality (1.8), and
K denote the familiar class of univalent and convex functions in U. Then for every
g ∈ K, we have
(1+λ)[1+β(2γ−1)]b2
[(1+λ)[1+β(2γ−1)]b2 +2βγ(1−α)] (f
∗ φ)(z) ≺ φ(z),
(z ∈ U, bk ≥ b2 > 0(k ≥ 2); γ ∈ C\ {0} ; 0 ≤ α < 1)
(2.1)
[(1 + λ)[1 + β(2γ − 1)]b2 + 2βγ(1 − α)]
2(1 + λ)[1 + β(2γ − 1)]b2
(2.2)
and
Re {f (z)} > −
(z ∈ U)
(1+λ)[1+β(2γ−1)]b2
The following constant factor 2[(1+λ)[1+β(2γ−1)]b
, in the subordination re2 +2βγ(1−α)]
sult (2.1) is the best dominant.
Proof. Let f (z) satisfy the inequality and let φ(z) = z +
∞
P
ck z k ∈ K, then
k=2
(1 + λ)[1 + β(2γ − 1)]b2
(f ∗ φ)(z)
2[(1 + λ)[1 + β(2γ − 1)]b2 + 2βγ(1 − α)]
51
(2.3)
V. G. Gupta, B. Shrama - Subordination results for certain classes of ...
∞
X
(1 + λ)[1 + β(2γ − 1)]b2
(z +
ak ck z k ).
2[(1 + λ)[1 + β(2γ − 1)]b2 + 2βγ(1 − α)]
k=2
=
By invoking definition (2.2), the subordination (2.1) our theorem will hold true if
the sequence
(1 + λ)[1 + β(2γ − 1)]b2
ak
2[(1 + λ)[1 + β(2γ − 1)]b2 + 2βγ(1 − α)]
∞
(2.4)
k=1
is a subordination factor sequence. By virtue of Lemma (1.1), this is equivalent to
the inequality
∞
X
(
(1 + λ)[1 + β(2γ − 1)]b2
Re 1 +
ak z k
[(1
+
λ)[1
+
β(2γ
−
1)]b
+
2βγ(1
−
α)]
2
k=1
)
>0
(2.5)
Since bk ≥ b2 > 0 for k ≥ 2, we have
Re 1 +
≥1−
≥1−
∞
P
(1+λ)[1+β(2γ−1)]b2
k
[(1+λ)[1+β(2γ−1)]b2 +2βγ(1−α)] ak z
k=1
(1+λ)[1+β(2γ−1)]b2
[(1+λ)[1+β(2γ−1)]b2 +2βγ(1−α)] r
∞
P
1
(1
− (1+λ)[1+β(2γ−1)]b
+2βγ(1−α)]
2
k=2
(1+λ)[1+β(2γ−1)]b2
[(1+λ)[1+β(2γ−1)]b2 +2βγ(1−α)] r
∞
P
1
(1
− (1+λ)[1+β(2γ−1)]b
2 +2βγ(1−α)]
k=2
+ λ)[1 + β(2γ − 1)]b2 ak rk
+ λ(k − 1))[1 + β(2γ − 1)]bk ak rk .
By using Lemma 1.2, we can easily seen that
≥1−
(1+λ)[1+β(2γ−1)]b2
[(1+λ)[1+β(2γ−1)]b2 +2βγ(1−α)] r
2βγ(1−α)
− [(1+λ)[1+β(2γ−1)]b
r
2 +2βγ(1−α)]
> 0,
(|z| = r < 1).
This establishes the inequality (2.5), and consequently the subordination relation
(2.1) of Theorem 2.1 is proved. The inequality (2.2) follows from (2.1), upon setting
∞
X
z
φ(z) =
=
zn ∈ K
1 − z n=1
and
f0 (z) = z −
(z ∈ U),
2βγ(1 − α)
z2
[(1 + λ)[1 + β(2γ − 1)]b2 + 2βγ(1 − α)]
(2.6)
(z ∈ U),
(2.7)
which belongs to PTg (λ, α, β, γ). Using (2.1), we infer that
(1 + λ)[1 + β(2γ − 1)]b2
z
f0 (z) ≺
[(1 + λ)[1 + β(2γ − 1)]b2 + 2βγ(1 − α)]
1−z
52
(2.8)
V. G. Gupta, B. Shrama - Subordination results for certain classes of ...
It can be easily verified for the function f0 (z) defined by (2.7) that
M in Re
|z| −
[(1 + λ)[1 + β(2γ − 1)]b2 + 2βγ(1 − α)]
2
(2.9)
which completes the proof of theorem.
By substituting λ = 0 , in Theorem (2.1), we easily get
Corollary 2.2.Let the function f (z) ∈ PTg (0, α, β, γ) satisfy the inequality
∞
X
[1 + β(2γ − 1)]ak bk ≤ 2βγ(1 − α)
(2.10)
k=2
and K denote the familiar class of univalent and convex functions in U. Then for
every φ ∈ K , we have
[1+β(2γ−1)]b2
[1+β(2γ−1)]b2 +2βγ(1−α)] (f
∗ φ)(z) ≺ φ(z)
(z ∈ U, bk ≥ b2 > 0(k ≥ 2); γ ∈ C/ {0} ; 0 ≤ α < 1)
and
Re {f (z)} > −
[1 + β(2γ − 1)]b2 + 2βγ(1 − α)]
2[1 + β(2γ − 1)]b2
(2.11)
(2.12)
[1+β(2γ−1)]b2 ]
, in the subordination result
The following constant factor 2[1+β(2γ−1)]b
2 +2βγ(1−α)
z
(2.11) is the best dominant. By setting φ(z) = 1−z
,β = 1 and λ = 0 ,
Corollary 2.3.Let the function f (z) ∈ PTg (0, α, 1, γ) satisfy the inequality
∞
X
2γak ≤ 2(1 − α)
(2.13)
k=2
and K denote the familiar class of univalent and convex functions in U. Then for
every φ(z) ∈ K, we have
1
2−α (f
∗ φ)(z) ≺ φ(z)
(z ∈ U, bk ≥ b2 > 0(k ≥ 2); γ ∈ C/ {0} ; 0 ≤ α < 1)
and
Re {f (z)} >
α−2
2
(2.14)
(2.15)
2
The following constant factor α−2
, in the subordination result (2.14) is the best
dominant.
3. Further remarks and observation
53
V. G. Gupta, B. Shrama - Subordination results for certain classes of ...
Using Hadamard Product (or convolution) defined by (1.2) and using Wilf lemma,
we obtained the subordination results for the subclass PTg (λ, α, β, γ) of A. If we
replace g (z) in Theorem 2.1 defined by the functions in (1.5) and (1.6) , we get the
corresponding results of the Theorem 2.1.
References
[1] N. E. Cho, H. M. Srivastava, Argument estimates of certain analytic functions
defined by a class of multiplier transformations, Math. Comput. Modelling,37, 12(2003), 39-49.
[2] J. Dziok, H. M. Srivastava, Classes of analytic functions associated with the
generalized hypergeometric function, Appl. Math. Comput., 103, 1 (1999), 1-13
[3] J. Dziok, H. M. Srivastava, Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function, Adv.
Stud. Contemp. Math , 5 (2002), 115-125.
[4] J. Dziok, H. M. Srivastava, Certain subclasses of analytic functions associated
with the generalized hypergeometric function, Integral Transforms Spec. Funct., 14
(2003), 7-18.
[5] G. Murugusundaramoorthy, S.B. Joshi, Certain classes of analytic functions
with negative coefficients associated with a convolution structure, J. Math. Inq., 3(2)
(2009), 293-303.
[6] H. Silverman, Univalent functions with negative coefficients, Proc. Amer.
Math. Soc., 51 (1975), 109-116.
[7] H. S. Wilf, Subordinating factor sequence for convex maps of the unit circle,
Proc. Amer. Math. Soc., 12 (1961), 689-693.
V. G. Gupta
Department of Mathematics
University of Rajasthan
Jaipur (INDIA)- 302055
email:[email protected]
Bhavna Sharma
Department of Mathematics
University Rajasthan
Jaipur (INDIA) - 302055
email:vallabhi [email protected]
54
No 20/2009
SUBORDINATION RESULTS FOR CERTAIN CLASSES OF
ANALYTIC FUNCTIONS ASSOCIATED WITH THE
CONVOLUTION STRUCTURE
V. G. Gupta and Bhavna Shrama
Abstract. The main object of the present paper is to derive subordination
property of the class which was very recently PTg (λ, α, β, γ) introduced by Murugusundaramoorthy and Joshi (J. Math. Ineq., 3(1)(2009), pp. 293-303).
2000 Mathematics Subject Classification: 30C45.
Keywords: Univalent functions, Convolution, Subordination .
1. Introduction and Preliminaries
Let A be the class of analytic functions of the form
f (z) = z +
∞
X
ak z k
(z ∈ ∆ := {z ∈ C| |z| < 1})
(1.1)
k=2
which are analytic and univalent in the open unit disk. If U = {z ∈ C : |z| < 1} is
given by (1.1) and g ∈ A is given by
g(z) = z +
∞
X
bk z k
(1.2)
k=2
the Hadamard product (or convolution ) (f ∗ g) of f and g is given by
(f ∗ g)(z) = z +
∞
X
k=2
49
ak bk z k , z ∈ U
(1.3)
V. G. Gupta, B. Shrama - Subordination results for certain classes of ...
Very recently, by using Hadamard product (or convolution) Murugusundaramoorthy
and Joshi [5], introduced the subclass Pg (λ, α, β, γ) of A consisting the functions of
the form (1.1) and satisfying the following inequality,
Jg,λ (z) − 1
< β, λ ≥ 0, 0 ≤ α < 1, 0 0
(z ∈ U)
(1.8)
k=1
To prove our main result, we shall required the following lemma due to Murusundaramoorthy and Joshi.
Lemma 1.2 (see [5]). Let the function f be defined by (1.4) then PTg (λ, α, β, γ) if
and only if
∞
X
(1 + λ(k − 1))[1 + β(2γ − 1)]ak bk ≤ 2βγ(1 − α)
(1.9)
k=2
2. Subordination theorem
Theorem 2.1.Let the function f ∈ PTg (λ, α, β, γ) satisfy the inequality (1.8), and
K denote the familiar class of univalent and convex functions in U. Then for every
g ∈ K, we have
(1+λ)[1+β(2γ−1)]b2
[(1+λ)[1+β(2γ−1)]b2 +2βγ(1−α)] (f
∗ φ)(z) ≺ φ(z),
(z ∈ U, bk ≥ b2 > 0(k ≥ 2); γ ∈ C\ {0} ; 0 ≤ α < 1)
(2.1)
[(1 + λ)[1 + β(2γ − 1)]b2 + 2βγ(1 − α)]
2(1 + λ)[1 + β(2γ − 1)]b2
(2.2)
and
Re {f (z)} > −
(z ∈ U)
(1+λ)[1+β(2γ−1)]b2
The following constant factor 2[(1+λ)[1+β(2γ−1)]b
, in the subordination re2 +2βγ(1−α)]
sult (2.1) is the best dominant.
Proof. Let f (z) satisfy the inequality and let φ(z) = z +
∞
P
ck z k ∈ K, then
k=2
(1 + λ)[1 + β(2γ − 1)]b2
(f ∗ φ)(z)
2[(1 + λ)[1 + β(2γ − 1)]b2 + 2βγ(1 − α)]
51
(2.3)
V. G. Gupta, B. Shrama - Subordination results for certain classes of ...
∞
X
(1 + λ)[1 + β(2γ − 1)]b2
(z +
ak ck z k ).
2[(1 + λ)[1 + β(2γ − 1)]b2 + 2βγ(1 − α)]
k=2
=
By invoking definition (2.2), the subordination (2.1) our theorem will hold true if
the sequence
(1 + λ)[1 + β(2γ − 1)]b2
ak
2[(1 + λ)[1 + β(2γ − 1)]b2 + 2βγ(1 − α)]
∞
(2.4)
k=1
is a subordination factor sequence. By virtue of Lemma (1.1), this is equivalent to
the inequality
∞
X
(
(1 + λ)[1 + β(2γ − 1)]b2
Re 1 +
ak z k
[(1
+
λ)[1
+
β(2γ
−
1)]b
+
2βγ(1
−
α)]
2
k=1
)
>0
(2.5)
Since bk ≥ b2 > 0 for k ≥ 2, we have
Re 1 +
≥1−
≥1−
∞
P
(1+λ)[1+β(2γ−1)]b2
k
[(1+λ)[1+β(2γ−1)]b2 +2βγ(1−α)] ak z
k=1
(1+λ)[1+β(2γ−1)]b2
[(1+λ)[1+β(2γ−1)]b2 +2βγ(1−α)] r
∞
P
1
(1
− (1+λ)[1+β(2γ−1)]b
+2βγ(1−α)]
2
k=2
(1+λ)[1+β(2γ−1)]b2
[(1+λ)[1+β(2γ−1)]b2 +2βγ(1−α)] r
∞
P
1
(1
− (1+λ)[1+β(2γ−1)]b
2 +2βγ(1−α)]
k=2
+ λ)[1 + β(2γ − 1)]b2 ak rk
+ λ(k − 1))[1 + β(2γ − 1)]bk ak rk .
By using Lemma 1.2, we can easily seen that
≥1−
(1+λ)[1+β(2γ−1)]b2
[(1+λ)[1+β(2γ−1)]b2 +2βγ(1−α)] r
2βγ(1−α)
− [(1+λ)[1+β(2γ−1)]b
r
2 +2βγ(1−α)]
> 0,
(|z| = r < 1).
This establishes the inequality (2.5), and consequently the subordination relation
(2.1) of Theorem 2.1 is proved. The inequality (2.2) follows from (2.1), upon setting
∞
X
z
φ(z) =
=
zn ∈ K
1 − z n=1
and
f0 (z) = z −
(z ∈ U),
2βγ(1 − α)
z2
[(1 + λ)[1 + β(2γ − 1)]b2 + 2βγ(1 − α)]
(2.6)
(z ∈ U),
(2.7)
which belongs to PTg (λ, α, β, γ). Using (2.1), we infer that
(1 + λ)[1 + β(2γ − 1)]b2
z
f0 (z) ≺
[(1 + λ)[1 + β(2γ − 1)]b2 + 2βγ(1 − α)]
1−z
52
(2.8)
V. G. Gupta, B. Shrama - Subordination results for certain classes of ...
It can be easily verified for the function f0 (z) defined by (2.7) that
M in Re
|z| −
[(1 + λ)[1 + β(2γ − 1)]b2 + 2βγ(1 − α)]
2
(2.9)
which completes the proof of theorem.
By substituting λ = 0 , in Theorem (2.1), we easily get
Corollary 2.2.Let the function f (z) ∈ PTg (0, α, β, γ) satisfy the inequality
∞
X
[1 + β(2γ − 1)]ak bk ≤ 2βγ(1 − α)
(2.10)
k=2
and K denote the familiar class of univalent and convex functions in U. Then for
every φ ∈ K , we have
[1+β(2γ−1)]b2
[1+β(2γ−1)]b2 +2βγ(1−α)] (f
∗ φ)(z) ≺ φ(z)
(z ∈ U, bk ≥ b2 > 0(k ≥ 2); γ ∈ C/ {0} ; 0 ≤ α < 1)
and
Re {f (z)} > −
[1 + β(2γ − 1)]b2 + 2βγ(1 − α)]
2[1 + β(2γ − 1)]b2
(2.11)
(2.12)
[1+β(2γ−1)]b2 ]
, in the subordination result
The following constant factor 2[1+β(2γ−1)]b
2 +2βγ(1−α)
z
(2.11) is the best dominant. By setting φ(z) = 1−z
,β = 1 and λ = 0 ,
Corollary 2.3.Let the function f (z) ∈ PTg (0, α, 1, γ) satisfy the inequality
∞
X
2γak ≤ 2(1 − α)
(2.13)
k=2
and K denote the familiar class of univalent and convex functions in U. Then for
every φ(z) ∈ K, we have
1
2−α (f
∗ φ)(z) ≺ φ(z)
(z ∈ U, bk ≥ b2 > 0(k ≥ 2); γ ∈ C/ {0} ; 0 ≤ α < 1)
and
Re {f (z)} >
α−2
2
(2.14)
(2.15)
2
The following constant factor α−2
, in the subordination result (2.14) is the best
dominant.
3. Further remarks and observation
53
V. G. Gupta, B. Shrama - Subordination results for certain classes of ...
Using Hadamard Product (or convolution) defined by (1.2) and using Wilf lemma,
we obtained the subordination results for the subclass PTg (λ, α, β, γ) of A. If we
replace g (z) in Theorem 2.1 defined by the functions in (1.5) and (1.6) , we get the
corresponding results of the Theorem 2.1.
References
[1] N. E. Cho, H. M. Srivastava, Argument estimates of certain analytic functions
defined by a class of multiplier transformations, Math. Comput. Modelling,37, 12(2003), 39-49.
[2] J. Dziok, H. M. Srivastava, Classes of analytic functions associated with the
generalized hypergeometric function, Appl. Math. Comput., 103, 1 (1999), 1-13
[3] J. Dziok, H. M. Srivastava, Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function, Adv.
Stud. Contemp. Math , 5 (2002), 115-125.
[4] J. Dziok, H. M. Srivastava, Certain subclasses of analytic functions associated
with the generalized hypergeometric function, Integral Transforms Spec. Funct., 14
(2003), 7-18.
[5] G. Murugusundaramoorthy, S.B. Joshi, Certain classes of analytic functions
with negative coefficients associated with a convolution structure, J. Math. Inq., 3(2)
(2009), 293-303.
[6] H. Silverman, Univalent functions with negative coefficients, Proc. Amer.
Math. Soc., 51 (1975), 109-116.
[7] H. S. Wilf, Subordinating factor sequence for convex maps of the unit circle,
Proc. Amer. Math. Soc., 12 (1961), 689-693.
V. G. Gupta
Department of Mathematics
University of Rajasthan
Jaipur (INDIA)- 302055
email:[email protected]
Bhavna Sharma
Department of Mathematics
University Rajasthan
Jaipur (INDIA) - 302055
email:vallabhi [email protected]
54