Journal of Inequalities in Pure and
Applied Mathematics
THE DZIOK-SRIVASTAVA OPERATOR AND k−UNIFORMLY
STARLIKE FUNCTIONS
volume 6, issue 2, article 52,
2005.
R. AGHALARY AND GH. AZADI
University of Urmia
Urmia, Iran
Received 04 January, 2005;
accepted 12 April, 2005.
Communicated by: H.M. Srivastava
EMail: raghalary@yahoo.com
EMail: azadi435@yahoo.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
003-05
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Abstract
Inclusion relations for k−uniformly starlike functions under the Dziok-Srivastava
operator are established. These results are also extended to k−uniformly convex functions, close-to-convex, and quasi-convex functions.
2000 Mathematics Subject Classification: Primary 30C45; Secondary 30C50.
Key words: Starlike, Convex, Linear operators.
The Dziok-Srivastava Operator
and k−Uniformly Starlike
Functions
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
7
R. Aghalary and Gh. Azadi
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1.
Introduction
P
n
Let A denote the class of functions of the form f (z) = z + ∞
n=2 an z which
are analytic in the open unit disc U = {z : |z| < 1}. A function f ∈ A is
said to be in U ST (k, γ), the class of k−uniformly starlike functions of order
γ, 0 ≤ γ < 1, if f satisfies the condition
0
0 zf (z)
zf (z)
(1.1)
<
> k − 1 + γ,
k ≥ 0.
f (z)
f (z)
Replacing f in (1.1) by zf 0 we obtain the condition
00 zf (z) zf 00 (z)
+ γ,
(1.2)
< 1+ 0
> k 0
f (z)
f (z) The Dziok-Srivastava Operator
and k−Uniformly Starlike
Functions
k≥0
required for the function f to be in the subclass U CV (k, γ) of k−uniformaly
convex functions of order γ.
Uniformly starlike and convex functions were first introduced by Goodman
[5] and then studied by various authors. For a wealth of references, see Ronning
[13].
Setting
o
n
p
Ωk,γ = u + iv; u > k (u − 1)2 + v 2 + γ ,
zf 0 (z)
f (z)
zf 00 (z)
f 0 (z)
with p(z) =
or p(z) = 1 +
and considering the functions which
map U on to the conic domain Ωk,γ , such that 1 ∈ Ωk,γ , we may rewrite the
conditions (1.1) or (1.2) in the form
(1.3)
p(z) ≺ qk,γ (z).
R. Aghalary and Gh. Azadi
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We note that the explicit forms of function qk,γ for k = 0 and k = 1 are
1 + (1 − 2γ)z
2(1 − γ)
q0,γ (z) =
, and q1,γ (z) = 1 +
1−z
π2
√ 2
1+ z
√
.
log
1− z
For 0 < k < 1 we obtain
1−γ
qk,γ (z) =
cos
1 − k2
√ 2
1+ z
k2 − γ
√
(arccos k)i log
−
,
π
1 − k2
1− z
The Dziok-Srivastava Operator
and k−Uniformly Starlike
Functions
and if k > 1, then qk,γ has the form
1−γ
qk,γ (z) = 2
sin
k −1
π
2K(k)
Z
0
u(z)
√
k
p
dt
√
1 − t2 1 − k 2 t2
√
z−√ k
1− kz
!
k2 − γ
+ 2
,
k −1
πK 0 (z)
.
4K(z)
where u(z) =
and K is such that k = cosh
By virtue of (1.3) and the properties of the domains Ωk,γ we have
(1.4)
<(p(z)) > <(qk,γ (z)) >
k+γ
.
k+1
Define U CC(k, γ, β) to be the family of functions f ∈ A such that
0
0 zf (z)
zf (z)
≥ k − 1 + γ, k≥0, 0≤γ < 1
<
g(z)
g(z)
for some g ∈ U ST (k, β).
R. Aghalary and Gh. Azadi
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Similarly, we define U QC(k, γ, β) to be the family of functions f ∈ A such
that
(zf 0 (z))0
(zf 0 (z))0
<
≥ k 0
− 1 + γ,
k ≥ 0, 0 ≤ γ < 1
0
g (z)
g (z)
for some g ∈ U CV (k, β).
We note that U CC(0, γ, β) is the class of close-to-convex functions of order
γ and type β and U QC(0, γ, β) is the class of quasi-convex functions of order
γ and type β.
The aim of this paper is to study the inclusion properties of the above mentioned classes under the following linear operator which is defined by Dziok
and Srivastava [3].
For αj ∈ C (j = 1, 2, 3, . . . , l) and βj ∈ C − {0, −1, −2, . . . } (j =
1, 2, . . . m), the generalized hypergeometric function is defined by
The Dziok-Srivastava Operator
and k−Uniformly Starlike
Functions
R. Aghalary and Gh. Azadi
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∞
X
(α1 )n · · · (αl )n z n
F
(α
,
.
.
.
,
α
;
β
,
.
.
.
,
β
)
=
· ,
l m
1
l
1
m
(β
)
·
·
·
(β
)
n!
1
n
m
n
n=0
JJ
J
(l ≤ m + 1; l, m ∈ N0 = {0, 1, 2, . . . }),
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where (a)n is the Pochhammer symbol defined by (a)n =
1) · · · (a + n − 1) for n ∈ N = {1, 2, . . . } and 1 when n = 0.
Corresponding to the function
Γ(a+n)
Γ(a)
= a(a +
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h(α1 , . . . , αl ; β1 , . . . , βm ; z) = z l Fm (α1 , . . . , αl ; β1 , . . . , βm )
J. Ineq. Pure and Appl. Math. 6(2) Art. 52, 2005
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l
the Dziok-Srivastava operator [3], Hm
(α1 , . . . , αl ; β1 , . . . , βm ) is defined by
l
Hm
(α1 , . . . , αl ; β1 , . . . , βm )f (z) = h(α1 , . . . , αl ; β1 , . . . , βm ; z) ∗ f (z)
∞
X
(α1 )n−1 · · · (αl )n−1
an z n
=z+
·
(β1 )n−1 · · · (βm )n−1 (n − 1)!.
n=2
where “∗” stands for convolution.
It is well known [3] that
l
(1.5) α1 Hm
(α1 + 1, . . . , αl ; β1 , . . . , βm )f (z)
l
= z[Hm
(α1 , . . . , αl ; β1 , . . . , βm ; z)f (z)]0
+ (α1 −
The Dziok-Srivastava Operator
and k−Uniformly Starlike
Functions
R. Aghalary and Gh. Azadi
l
1)Hm
(α1 , . . . , αl ; β1 , . . . , βm )f (z).
To make the notation simple, we write,
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l
[α1 ]f (z)
Hm
=
l
Hm
(α1 , . . . , αl ; β1 , . . . , βm ; z)f (z).
We note that many subclasses of analytic functions, associated with the
l
Dziok-Srivastava operator Hm
[α1 ] and many special cases, were investigated
recently by Dziok-Srivastava [3], Liu [7], Liu and Srivastava [9], [10] and others. Also we note that special cases of the Dziok-Srivastava linear operator
include the Hohlov linear operator [6], the Carlson-Shaffer operator [2], the
Ruscheweyh derivative operator [14], the generalized Bernardi-Libera-Livingston
linear operator (cf. [1]) and the Srivastava-Owa fractional derivative operators
(cf. [11], [12]).
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2.
Main Results
l
In this section we prove some results on the linear operator Hm
[α1 ]. First is the
inclusion theorem.
Theorem 2.1. Let <α1 >
l
Hm
[α1 ]f ∈ U ST (k, γ).
1−γ
,
k+1
l
and f ∈ A. If Hm
[α1 + 1]f ∈ U ST (k, γ) then
In order to prove the above theorem we shall need the following lemma
which is due to Eenigenburg, Miller, Mocanu, and Read [4].
Lemma A. Let β, γ be complex constants and h be univalently convex
P∞ in then
unit disk U with h(0) = c and <(βh(z) + γ) > 0. Let g(z) = c + n=1 pn z
be analytic in U . Then
g(z) +
zg 0 (z)
≺ h(z) ⇒ g(z) ≺ h(z).
βg(z) + γ
l
l
Proof of Theorem 2.1. Setting p(z) = z(Hm
[α1 ]f (z))0 /(Hm
[α1 ]f (z)) in (1.5)
we can write
l
l
Hm
[α1 + 1]f (z)
z(Hm
[α1 ]f (z))0
=
+ (α1 − 1) = p(z) + (α1 − 1).
(2.1) α1
l [α ]f (z)
l [α ]f (z)
Hm
Hm
1
1
Differentiating (2.1) yields
(2.2)
l
z(Hm
[α1 + 1]f (z))0
zp0 (z)
=
p(z)
+
.
l [α + 1]
Hm
p(z) + (α1 − 1)
1
The Dziok-Srivastava Operator
and k−Uniformly Starlike
Functions
R. Aghalary and Gh. Azadi
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From this and the argument given in Section 1 we may write
p(z) +
zp0 (z)
≺ qk,γ (z).
p(z) + (α1 − 1)
Therefore the theorem follows by Lemma A and the condition (1.4) since qk,γ
is univalent and convex in U and <(qk,γ ) > k+γ
.
k+1
Theorem 2.2. Let <α1 >
l
Hm
[α1 ]f ∈ U CV (k, γ).
1−γ
,
k+1
l
and f ∈ A. If Hm
[α1 + 1]f ∈ U CV (k, γ) then
Proof. By virtue of (1.1), (1.2) and Theorem 2.1 we have
0
l
l
Hm
[α1 + 1]f ∈ U CV (k, γ) ⇔ z Hm
[α1 + 1]f ∈ U ST (k, γ)
l
⇔ Hm
[α1 + 1]zf 0 ∈ U ST (k, γ)
⇒
⇔
l
Hm
[α1 ]zf 0 ∈ U ST (k, γ)
l
[α1 ]f ∈ U CV (k, γ).
Hm
and the proof is complete.
We next prove
The Dziok-Srivastava Operator
and k−Uniformly Starlike
Functions
R. Aghalary and Gh. Azadi
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l
Theorem 2.3. Let <α1 > 1−γ
, and f ∈ A. If Hm
[α1 + 1]f ∈ U CC(k, γ, β)
k+1
l
then Hm
[α1 ]f ∈ U CC(k, γ, β).
Close
To prove the above theorem, we shall need the following lemma which is
due to Miller and Mocanu [10].
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Lemma B. Let h be convex in the unit disk U and let E ≥ 0. Suppose B(z) is
analytic in U with <B(z) ≥ E. If g is analytic in U and g(0) = h(0). Then
Ez 2 g 00 (z) + B(z)zg 0 (z) + g(z) ≺ h(z) ⇒ g(z) ≺ h(z).
l
Proof of Theorem 2.3. Since Hm
[α1 + 1]f ∈ U CC(k, γ, β), by definition, we
can write
l
z(Hm
[α1 + 1]f )0 (z)
≺ qk,γ (z)
k(z)
l
for some k(z) ∈ U ST (k, β). For g such that Hm
[α1 + 1]g(z) = k(z), we have
l
z(Hm
[α1 + 1]f )0 (z)
≺ qk,γ (z).
l [α + 1]g(z)
Hm
1
(2.3)
0
l
l
0
z(Hm [α1 ]g) (z)
m [α1 ]f ) (z)
Letting h(z) = z(H
we observe that h and
l [α ]g)(z) and H(z) =
l [α ]g(z)
(Hm
Hm
1
1
l
H are analytic in U and h(0) = H(0) = 1. Now, by Theorem 2.1, Hm
[α1 ]g ∈
k+β
U ST (k, β) and so <H(z) > k+1 . Also, note that
(2.4)
l
l
z(Hm
[α1 ]f )0 (z) = (Hm
[α1 ]g(z))h(z).
The Dziok-Srivastava Operator
and k−Uniformly Starlike
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R. Aghalary and Gh. Azadi
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Differentiating both sides of (2.4) yields
l
z(Hm
[α1 ](zf 0 ))0 (z)
l [α ]g(z)
Hm
1
=
l
z(Hm
[α1 ]g)0 (z)
h(z)
l [α ]g(z)
Hm
1
Close
+ zh0 (z) = H(z)h(z) + zh0 (z).
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Now using the identity (1.5) we obtain
(2.5)
l
z(Hm
[α1 + 1]f )0 (z)
l [α + 1]g(z)
Hm
1
H l [α1 + 1](zf 0 )(z)
= ml
Hm [α1 + 1]g(z)
l
l
z(Hm
[α1 ](zf 0 ))0 (z) + (α1 − 1)Hm
[α1 ](zf 0 )(z)
=
l [α ]g)0 (z) + (α − 1)H l [α ]g(z)
z(Hm
1
1
m 1
=
=
l [α ](zf 0 ))0 (z)
l [α ](zf 0 )(z)
z(Hm
1
1
+ (α1 − 1) Hm
l [α ]g(z)
l [α ]g(z)
Hm
Hm
1
1
l [α ]g)0 (z)
z(Hm
1
+ (α1 − 1)
l [α ]g(z)
Hm
1
H(z)h(z) + zh0 (z) + (α1 − 1)h(z)
H(z) + (α1 − 1)
1
= h(z) +
zh0 (z).
H(z) + (α1 − 1)
From (2.3), (2.4), and (2.5) we conclude that
h(z) +
1
zh0 (z) ≺ qk,γ (z).
H(z) + (α1 − 1)
On letting E = 0 and B(z) =
<(B(z)) =
1
,
H(z)+(α1 −1)
we obtain
1
<((α1 − 1) + H(z)) > 0.
|(α1 − 1) + H(z)|2
The above inequality satisfies the conditions required by Lemma B. Hence
h(z) ≺ qkγ (z) and so the proof is complete.
The Dziok-Srivastava Operator
and k−Uniformly Starlike
Functions
R. Aghalary and Gh. Azadi
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Using a similar argument to that in Theorem 2.2 we can prove
l
Theorem 2.4. Let <α1 > 1−γ
, and f ∈ A. If Hm
[α1 + 1]f ∈ U QC(k, γ, β),
k+1
l
then Hm
[α1 ]f ∈ U QC(k, γ, β).
Finally, we examine the closure properties of the above classes of functions under the generalized Bernardi-Libera-Livingston integral operator Lc (f )
which is defined by
Z
c + 1 z c−1
t f (t)dt, c > −1.
Lc (f ) =
zc 0
Theorem 2.5. Let c >
−(k+γ)
.
k+1
l
l
If Hm
[α1 ]f ∈ U ST (k, γ) so is Lc (Hm
[α1 ]f ).
l
Proof. From definition of Lc (f ) and the linearity of operator Hm
[α1 ] we have
(2.6)
l
l
l
z(Hm
[α1 ]Lc (f ))0 (z) = (c + 1)Hm
[α1 ]f (z) − c(Hm
[α1 ]Lc (f ))(z).
Substituting
(2.7)
l [α ]L (f ))0 (z)
z(Hm
1 c
l [α ]L (f )(z)
Hm
1 c
= p(z) in (2.6) we may write
p(z) = (c + 1)
l
Hm
[α1 ]f (z)
− c.
l
(Hm [α1 ]Lc (f ))(z)
Differentiating (2.7) gives
l
z(Hm
[α1 ]f )0 (z)
zp0 (z)
=
p(z)
+
.
l [α ]f )(z)
(Hm
p(z) + c
1
Now, the theorem follows by Lemma A, since <(qk,γ (z) + c) > 0.
The Dziok-Srivastava Operator
and k−Uniformly Starlike
Functions
R. Aghalary and Gh. Azadi
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A similar argument leads to
Theorem 2.6. Let c >
−(k+γ)
.
k+1
l
l
If Hm
[α1 ]f ∈ U CV (k, γ) so is Lc (Hm
[α1 ]f ).
Theorem 2.7. Let c >
−(k+γ)
.
k+1
l
l
If Hm
[α1 ]f ∈ U CC(k, γ, β) so is Lc (Hm
[α1 ]f ).
l
Proof. By definition, there exists a function k(z) = (Hm
[α1 ]g)(z) ∈ U ST (k, β)
such that
l
z(Hm
[α1 ]f )0 (z)
(2.8)
≺ qk,γ (z)
(z ∈ U ).
l [α ]g)(z)
(Hm
1
Now from (2.6) we have
l
l
l
z(Hm
[α1 ]f )0 (z)
z(Hm
[α1 ]Lc (zf 0 ))0 (z) + cHm
[α1 ]Lc (zf 0 )(z)
=
l [α ]g)(z)
l [α ]L (g(z)))0 (z) + c(H l [α ]L (g))(z)
(Hm
z(Hm
1
1 c
m 1 c
(2.9)
l [α ]L (zf 0 ))0 (z)
l
0
z(Hm
1 c
m [α1 ]Lc (zf ))(z)
+ c(H
l [α ]L (g))(z)
l [α ]L (g))(z)
(Hm
(Hm
1 c
1 c
=
l [α ]L (g))0 (z)
z(Hm
1 c
l [α ]L (g))(z) + c
(Hm
1 c
.
l
l
Since Hm
[α1 ]g ∈ U ST (k, β), by Theorem 2.5, we have Lc (Hm
[α1 ]g) ∈ U ST (k, β).
l
0
z(Hm [α1 ]Lc (g))
k+β
Letting H l [α1 ]Lc (g) = H(z), we note that <(H(z)) > k+1 . Now, let h be
m
defined by
(2.10)
l
l
z(Hm
[α1 ]Lc (f ))0 = h(z)Hm
[α1 ]Lc (g).
Differentiating both sides of (2.10) yields
(2.11)
l
l
z(Hm
[α1 ](zLc (f ))0 )0 (z)
z(Hm
[α1 ]Lc (g))0 (z)
0
=
zh
(z)
+
h(z)
l [α ]L (g))(z)
l [α ]L (g))(z)
(Hm
(Hm
1 c
1 c
0
= zh (z) + H(z)h(z).
The Dziok-Srivastava Operator
and k−Uniformly Starlike
Functions
R. Aghalary and Gh. Azadi
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Therefore from (2.9) and (2.11) we obtain
l
z(Hm
[α1 ]f )0 (z)
zh0 (z) + h(z)H(z) + ch(z)
=
.
l [α ]g)(z)
(Hm
H(z) + c
1
This in conjunction with (2.8) leads to
zh0 (z)
h(z) +
≺ qk,γ (z).
H(z) + c
(2.12)
− k+β
.
k+1
1
H(z)+c
Letting B(z) =
in (2.12) we note that <(B(z)) > 0 if c >
Now for E = 0 and B as described we conclude the proof since the required
conditions of Lemma B are satisfied.
A similar argument yields
Theorem 2.8. Let c >
−(k+γ)
.
k+1
The Dziok-Srivastava Operator
and k−Uniformly Starlike
Functions
R. Aghalary and Gh. Azadi
Title Page
If
l
Hm
[α1 ]f
∈ U QC(k, γ, β) so is
l
Lc (Hm
[α1 ]f ).
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References
[1] S.D. BERNARDI, Convex and starlike univalent functions, Trans. Amer.
Math. Soc., 135 (1969), 429–446.
[2] B.C. CARLSON AND S.B. SHAFFER, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15 (1984), 737–745.
[3] J. DZIOK AND H.M. SRIVASTAVA, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral
Transform Spec.Funct., 14 (2003), 7–18.
[4] P. EEINGENBURG, S.S. MILLER, P.T. MOCANU AND M.D. READE,
General Inequalities, 64 (1983), (Birkhauseverlag-Basel) ISNM, 339–
348.
[5] A.W. GOODMAN, On uniformly starlike functions, J. Math. Anal. Appl.,
155 (1991), 364–370.
[6] Yu.E. HOHLOV, Operators and operations in the class of univalent functions, Izv. Vyss. Ucebn. Zaved. Mat., 10 (1978), 83–89.
[7] J.-L. LIU, Strongly starlike functions associated with the Dziok-Srivastava
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[8] J.-L. LIU AND H.M. SRIVASTAVA, Classes of meromorphically multivalent functions associated with the generalized hypergeometric function,
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