Volume 9 (2008), Issue 4, Article 102, 8 pp.
SUBORDINATION THEOREM FOR A FAMILY OF ANALYTIC FUNCTIONS
ASSOCIATED WITH THE CONVOLUTION STRUCTURE
J. K. PRAJAPAT
D EPARTMENT OF M ATHEMATICS
B HARTIYA I NSTITUTE OF E NGINEERING & T ECHNOLOGY
N EAR S ANWALI C IRCLE , B IKANER B Y-PASS ROAD
S IKAR -332001, R AJASTHAN , INDIA.
jkp_0007@rediffmail.com
Received 04 May, 2007; accepted 01 September, 2008
Communicated by S.S. Dragomir
A BSTRACT. We use the familiar convolution structure of analytic functions to introduce new
class of analytic functions of complex order. The results investigated in the present paper include,
the characterization and subordination properties for this class of analytic functions. Several
interesting consequences of our results are also pointed out.
Key words and phrases: Analytic function, Hadamard product(or convolution), Dziok-Srivastava linear operator, Subordination factor sequence, Characterization properties.
2000 Mathematics Subject Classification. 30C45.
1. I NTRODUCTION AND P RELIMINARIES
Let A denote the class of functions of the form
∞
X
(1.1)
f (z) = z +
ak z k ,
k=2
which are analytic and univalent in the open unit disk U = {z; z ∈ C : |z| < 1}. If f ∈ A is
given by (1.1) and g ∈ A is given by
∞
X
(1.2)
g(z) = z +
bk z k ,
k=2
then the Hadamard product (or convolution) f ∗ g of f and g is defined(as usual) by
∞
X
(1.3)
(f ∗ g)(z) := z +
ak b k z k .
k=2
In this article we study the class Sγ (g; α) introduced in the following:
The author expresses his sincerest thanks to the worthy referee for valuable comments. He is also thankful to Emeritus Professor Dr. R.K.
Raina for his useful suggestions.
139-07
2
J. K. P RAJAPAT
Definition 1.1. For a given function g(z) ∈ A defined by (1.2), where bk ≥ 0 (k ≥ 2). We say
that f (z) ∈ A is in Sγ (g; α), provided that (f ∗ g)(z) 6= 0, and
1 z(f ∗ g)0 (z)
−1
>α
(z ∈ U; γ ∈ C\{0}; 0 ≤ α < 1).
(1.4)
Re 1 +
γ
(f ∗ g)(z)
Note that
z
z
∗
; α = S (α) and S1
; α = K(α),
S1
1−z
(1 − z)2
are, respectively, the familiar classes of starlike and convex functions of order α in U (see, for
example, [11]). Also
z
z
∗
Sγ
; 0 = Sγ and Sγ
; 0 = Kγ ,
1−z
(1 − z)2
where the classes Sγ∗ and Kγ stem essentially from the classes of starlike and convex functions
of complex order, which were considered earlier by Nasr and Aouf [9] and Wiatrowski [12],
respectively (see also [7] and [8]).
Remark 1. When
g(z) = z +
(1.5)
∞
X
k=2
(α1 )k−1 · · · (αq )k−1
zk
(β1 )k−1 · · · (βs )k−1 (k − 1)!
(αj ∈ C(j = 1, 2, . . . , q), βj ∈ C\{0, −1, −2, . . . } (j = 1, 2, . . . , s)) ,
with the parameters
α1 , . . . , αq and β1 , . . . , βs ,
being so choosen that the coefficients bk in (1.2) satisfy the following condition:
(α1 )k−1 · · · (αq )k−1
(1.6)
bk =
≥ 0,
(β1 )k−1 · · · (βs )k−1 (k − 1)!
then the class Sγ (g; α) is transformed into a (presumbly) new class Sγ∗ (q, s, α) defined by
1 z(Hsq [α1 ]f )0 (z)
∗
(1.7)
Sγ (q, s, α) := f : f ∈ A and Re 1 +
−1
>α
γ
(Hsq [α1 ]f )(z)
(z ∈ U; q ≤ s + 1; q, s ∈ N0 ; γ ∈ C\{0}).
The operator
(Hsq [α1 ]f ) (z) := Hsq (α1 , . . . , αq ; β1 , . . . , βs )f (z),
involved in (1.7) is the Dziok-Srivastava linear operator (see for details, [3]) which contains such
well known operators as the Hohlov linear operator, Carlson-Shaffer linear operator, Ruscheweyh
derivative operator, the Barnardi-Libera-Livingston operator, and the Srivastava-Owa fractional
derivative operator. One may refer to the papers [3] to [5] for further details and references for
these operators. The Dziok-Srivastava linear operator defined in [3] was further extended by
Dziok and Raina [1] (see also [2]).
In our present investigation, we require the following definitions and a related result due to
Welf [13].
Definition 1.2 (Subordination Principal). For two functions f and g analytic in U, we say
that the function f (z) is subordinated to g(z) in U and write f (z) ≺ g(z) (z ∈ U), if there
exists a Schawarz function w(z) analytic in U with w(0) = 0, and |w(z)| < 1, such that
f (z) = g(w(z)), z ∈ U. In particular, if the function g(z) is univalent in U, the above
subordination is equivalent to f (0) = g(0) and f (U) ⊂ g(U).
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 102, 8 pp.
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S UBORDINATION T HEOREM FOR A FAMILY OF A NALYTIC F UNCTIONS
3
Definition 1.3 (Subordinating Factor Sequence). A sequence {bk }∞
k=1 of complex numbers is
called a subordinating factor sequence if, whenever
f (z) =
∞
X
ak z k
(a1 = 1),
k=1
is analytic, univalent and convex in U, we have the subordination given by
∞
X
(1.8)
ak bk z k ≺ f (z)
(z ∈ U).
k=1
Lemma 1.1 (Wilf, [13]). The sequence {bk }∞
k=1 is a subordinating factor sequence if and only
if
)
(
∞
X
bk z k > 0
(z ∈ U).
(1.9)
Re 1 + 2
k=1
2. C HARACTERIZATION P ROPERTIES
In this section we establish two results (Theorem 2.1 and Theorem 2.3) which give the sufficiency conditions for a function f (z) defined by (1.1) and belong to the class f (z) ∈ Sγ (g; α).
Theorem 2.1. Let f (z) ∈ A such that
z(f ∗ g)0 (z)
(2.1)
(f ∗ g)(z) − 1 < 1 − β
(β < 1; z ∈ U),
then f (z) ∈ Sγ (g; α), provided that
|γ| ≥
(2.2)
1−β
,
1−α
(0 ≤ α < 1).
Proof. In view of (2.1), we write
z(f ∗ g)0 (z)
= 1 + (1 − β)w(z) where |w(z)| < 1 for z ∈ U.
(f ∗ g)(z)
Now
1 z(f ∗ g)0 (z)
1
Re 1 +
−1
= Re 1 + (1 − β)w(z)
γ
(f ∗ g)(z)
γ
w(z)
= 1 + (1 − β) Re
γ
w(z) ≥ 1 − (1 − β) γ 1
> 1 − (1 − β) ·
|γ|
≥ α,
provided that |γ| ≥
1−β
.
1−α
This completes the proof.
If we set
β = 1 − (1 − α)|γ|
(0 ≤ α < 1; γ ∈ C\{0}),
in Theorem 2.1, we obtain
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 102, 8 pp.
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4
J. K. P RAJAPAT
Corollary 2.2. If f (z) ∈ A such that
z(f ∗ g)0 (z)
(2.3)
(f ∗ g)(z) − 1 < (1 − α)|γ|
(z ∈ U, 0 ≤ α < 1; γ ∈ C\{0}),
then f (z) ∈ Sγ (g; α).
Theorem 2.3. Let f (z) ∈ A satisfy the following inequality
∞
X
(2.4)
bk [(k − 1) + (1 − α)|γ| ]|ak | ≤ (1 − α)|γ|
k=2
(z ∈ U; bk ≥ 0 (k ≥ 2); γ ∈ C\{0}; 0 ≤ α < 1),
then f (z) ∈ Sγ (g; α).
Proof. Suppose the inequality (2.4) holds true. Then in view of Corollary 2.2, we have
|z(f ∗ g)0 (z) − (f ∗ g)(z)| − (1 − α)|γ| |(f ∗ g)(z)|
∞
∞
X
X
=
bk (k − 1)ak z k − (1 − α)|γ| z +
b k ak z k k=2
k=2
(∞
)
∞
X
X
≤
bk (k − 1)|ak | − (1 − α)|γ| + (1 − α)|γ|
bk |ak | |z|
≤
k=2
k=2
(∞
X
)
bk [(k − 1) + (1 − α)|γ| ]|ak | − (1 − α)|γ|
k=2
≤ 0.
This completes the proof.
On specializing the parameters, Theorem 2.1 would yield the following results:
Corollary 2.4. Let f (z) ∈ A satisfy the following inequality
∞
X
(2.5)
(k + |γ| − 1)|ak | ≤ |γ|
(z ∈ U, γ ∈ C\{0}),
k=2
then f (z) ∈ Sγ∗ .
Corollary 2.5. Let f (z) ∈ A satisfy the following inequality
∞
X
(2.6)
k(k + |γ| − 1)|ak | ≤ |γ|
(z ∈ U, γ ∈ C\{0}),
k=2
then f (z) ∈ Kγ .
Corollary 2.6. Let f (z) ∈ A satisfy the following inequality
(2.7)
∞
X
[(k − 1) + (1 − α)|γ|](α1 )k−1 · · · (αq )k−1
|ak | ≤ (1 − α)|γ|
(β1 )k−1 · · · (βs )k−1 (k − 1)!
k=2
(z ∈ U; q ≤ s + 1; q, s ∈ N0 ; γ ∈ C\{0}; 0 ≤ α < 1),
then f (z) ∈ Sγ∗ (q, s, α).
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S UBORDINATION T HEOREM FOR A FAMILY OF A NALYTIC F UNCTIONS
5
3. S UBORDINATION T HEOREM
Theorem 3.1. Let the function f (z) ∈ A satisfy the inequality (2.4), and K denote the familiar
class of functions h(z) ∈ A which are univalent and convex in U. Then for every ψ ∈ K, we
have
[1 + (1 − α)|γ|]b2
(f ∗ ψ)(z) ≺ ψ(z)
2[b2 + (1 − α)(b2 + 1)|γ|]
(z ∈ U; bk ≥ b2 > 0 (k ≥ 2); γ ∈ C\{0}; 0 ≤ α < 1),
(3.1)
and
Re{f (z)} > −
(3.2)
[b2 + (1 − α)(b2 + 1)|γ|]
[1 + (1 − α)|γ|]b2
(z ∈ U).
The following constant factor
[1 + (1 − α)|γ|]b2
2[b2 + (1 − α)(b2 + 1)|γ|]
in the subordination result (3.1) is the best dominant.
Proof. Let f (z) satisfy the inequality (2.4) and let ψ(z) =
P∞
k=0 ck z
k+1
∈ K, then
[1 + (1 − α)|γ|]b2
(f ∗ ψ)(z)
2[b2 + (1 − α)(b2 + 1)|γ|]
(3.3)
[1 + (1 − α)|γ|]b2
=
2[b2 + (1 − α)(b2 + 1)|γ|]
z+
∞
X
!
ak c k z k
.
k=2
By invoking Definition 1.3, the subordination (3.1) of our theorem will hold true if the sequence
(3.4)
[1 + (1 − α)|γ|]b2
ak
2[b2 + (1 − α)(b2 + 1)|γ|]
∞
,
k=1
is a subordination factor sequence. By virtue of Lemma 1.1, this is equivalent to the inequality
(
)
∞
X
[1 + (1 − α)|γ|]b2
(3.5)
Re 1 +
ak z k > 0
(z ∈ U).
[b
2 + (1 − α)(b2 + 1)|γ|]
k=1
Since bk ≥ b2 > 0 for k ≥ 2, we have
(
Re 1 +
∞
X
k=1
[1 + (1 − α)|γ|]b2
ak z k
[b2 + (1 − α)(b2 + 1)|γ|]
)
∞
(
X
[1 + (1 − α)|γ|]b2
1
= Re 1 +
z+
[1 + (1 − α)|γ|]b2 ak z k
[b2 + (1 − α)(b2 + 1)|γ|]
[b2 + (1 − α)(b2 + 1)|γ|]
)
k=2
≥1−
>1−
[1 + (1 − α)|γ|]b2
1
r−
[b2 + (1 − α)(b2 + 1)|γ|]
[b2 + (1 − α)(b2 + 1)|γ|]
∞
X
[(k − 1) + (1 − α)|γ|]bk |ak |rk
k=2
[1 + (1 − α)|γ|]b2
(1 − α)|γ|
r−
r>0
[b2 + (1 − α)(b2 + 1)|γ|]
[b2 + (1 − α)(b2 + 1)|γ|]
(|z| = r < 1).
This establishes the inequality (3.5), and consequently the subordination relation (3.1) of Theorem 3.1 is proved. The assertion (3.2) follows readily from (3.1) when the function ψ(z) is
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 102, 8 pp.
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6
J. K. P RAJAPAT
selected as
∞
X
z
ψ(z) =
=z+
z k ∈ K.
1−z
k=2
(3.6)
The sharpness of the multiplying factor in (3.1) can be established by considering a function
h(z) defined by
h(z) = z −
(1 − α)|γ|
z2
[1 + (1 − α)|γ|]
(z ∈ U; γ ∈ C\{0}; 0 ≤ α < 1),
which belongs to the class Sγ (g; α). Using (3.1), we infer that
[1 + (1 − α)|γ|]b2
z
h(z) ≺
.
2[b2 + (1 − α)(b2 + 1)|γ|]
1−z
It can easily be verified that
[1 + (1 − α)|γ|]b2
1
h(z) = − ,
(3.7)
min
|z|≤1 2[b2 + (1 − α)(b2 + 1)|γ|]
2
which shows that the constant
[1 + (1 − α)|γ|]b2
2[b2 + (1 − α)(b2 + 1)|γ|]
is the best estimate.
Before concluding this paper, we consider some useful consequences of the subordination
Theorem 3.1.
Corollary 3.2. Let the function f (z) defined by (1.1) satisfy the inequality (2.5). Then for every
ψ ∈ K, we have
(1 + |γ|)
(f ∗ ψ)(z) ≺ ψ(z)
(z ∈ U),
(3.8)
2(1 + 2|γ|)
and
(1 + 2|γ|)
(3.9)
Re{f (z)} > −
(z ∈ U).
(1 + |γ|)
The constant factor
(1 + |γ|)
,
2(1 + 2|γ|)
in the subordination result (3.8) is the best dominant.
Corollary 3.3. Let the function f (z) defined by (1.1) satisfy the inequality (2.6). Then for every
ψ ∈ K, we have
(1 + |γ|)
(3.10)
(f ∗ ψ)(z) ≺ ψ(z)
(z ∈ U),
(2 + 3|γ|)
and
2(2 + 3|γ|)
(3.11)
Re{f (z)} > −
(z ∈ U).
(1 + |γ|)
The constant factor
(1 + |γ|)
,
(2 + 3|γ|)
in the subordination result (3.10) is the best dominant.
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 102, 8 pp.
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S UBORDINATION T HEOREM FOR A FAMILY OF A NALYTIC F UNCTIONS
7
Corollary 3.4. Let the function f (z) defined by (1.1) satisfy the inequality (2.7). Then for every
ψ ∈ K, we have
(3.12)
[1 + (1 − α)|γ|]c2
(f ∗ ψ)(z) ≺ ψ(z)
2[c2 + (1 − α)(c2 + 1)|γ|]
(z ∈ U),
and
(3.13)
Re{f (z)} > −
[c2 + (1 − α)(c2 + 1)|γ|]
[1 + (1 − α)|γ|]c2
(z ∈ U).
The constant factor
[1 + (1 − α)|γ|]c2
,
2[c2 + (1 − α)(c2 + 1)|γ|]
in the subordination result (3.12) is the best dominant, where c2 is given by
α1 · · · αq
.
c2 =
β1 · · · βs
Remark 2. On setting γ = 1 in Corollaries 3.2 and 3.3, we obtain results that correspond to
those of Frasin [6, p. 5, Corollary 2.4; p. 6 , Corollary 2.7] (see also, Singh [10, p. 434,
Corollary 2.2]).
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J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 102, 8 pp.
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