Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 2, Article 52, 2005
THE DZIOK-SRIVASTAVA OPERATOR AND k−UNIFORMLY STARLIKE
FUNCTIONS
R. AGHALARY AND GH. AZADI
U NIVERSITY OF U RMIA
U RMIA , I RAN
raghalary@yahoo.com
azadi435@yahoo.com
Received 04 January, 2005; accepted 12 April, 2005
Communicated by H.M. Srivastava
A BSTRACT. 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.
Key words and phrases: Starlike, Convex, Linear operators.
2000 Mathematics Subject Classification. Primary 30C45; Secondary 30C50.
1. I NTRODUCTION
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 zf (z)
zf 0 (z)
(1.1)
<
>k
− 1 + γ,
k ≥ 0.
f (z)
f (z)
Replacing f in (1.1) by zf 0 we obtain the condition
zf 00 (z)
zf 00 (z)
(1.2)
< 1+ 0
>k
+ γ,
f (z)
f 0 (z)
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
n
o
p
2
2
Ωk,γ = u + iv; u > k (u − 1) + v + γ ,
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
003-05
2
R. AGHALARY AND G H . A ZADI
0
00
(z)
(z)
or p(z) = 1 + zff 0 (z)
and considering the functions which map U on to the
with p(z) = zff (z)
conic domain Ωk,γ , such that 1 ∈ Ωk,γ , we may rewrite the conditions (1.1) or (1.2) in the form
p(z) ≺ qk,γ (z).
(1.3)
We note that the explicit forms of function qk,γ for k = 0 and k = 1 are
√ 2
1+ z
2(1 − γ)
1 + (1 − 2γ)z
√
.
, and q1,γ (z) = 1 +
log
q0,γ (z) =
1−z
π2
1− z
For 0 < k < 1 we obtain
1−γ
cos
qk,γ (z) =
1 − k2
√ k2 − γ
2
1+ z
√
,
−
(arccos k)i log
1 − k2
π
1− z
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
!
dt
√
p
+
1 − t2 1 − k 2 t2
k2 − γ
,
k2 − 1
0
(z)
z−√ k
where u(z) = 1−
and K is such that k = cosh πK
.
4K(z)
kz
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 zf (z)
zf 0 (z)
<
≥k
− 1 + γ, k≥0, 0≤γ < 1
g(z)
g(z)
for some g ∈ U ST (k, β).
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
− 1 + γ,
k ≥ 0, 0 ≤ γ < 1
g 0 (z)
g 0 (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
∞
X
(α1 )n · · · (αl )n z n
· ,
l Fm (α1 , . . . , αl ; β1 , . . . , βm ) =
(β1 )n · · · (βm )n n!
n=0
(l ≤ m + 1; l, m ∈ N0 = {0, 1, 2, . . . }),
where (a)n is the Pochhammer symbol defined by (a)n =
n ∈ N = {1, 2, . . . } and 1 when n = 0.
J. Inequal. Pure and Appl. Math., 6(2) Art. 52, 2005
Γ(a+n)
Γ(a)
= a(a + 1) · · · (a + n − 1) for
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T HE D ZIOK -S RIVASTAVA O PERATOR AND k−U NIFORMLY S TARLIKE F UNCTIONS
3
Corresponding to the function h(α1 , . . . , αl ; β1 , . . . , βm ; z) = z l Fm (α1 , . . . , αl ; β1 , . . . , βm )
l
the Dziok-Srivastava operator [3], Hm
(α1 , . . . , αl ; β1 , . . . , βm ) is defined by
l
(α1 , . . . , αl ; β1 , . . . , βm )f (z) = h(α1 , . . . , αl ; β1 , . . . , βm ; z) ∗ f (z)
Hm
∞
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
l
+ (α1 − 1)Hm
(α1 , . . . , αl ; β1 , . . . , βm )f (z).
To make the notation simple, we write,
l
l
(α1 , . . . , αl ; β1 , . . . , βm ; z)f (z).
Hm
[α1 ]f (z) = Hm
We note that many subclasses of analytic functions, associated with the Dziok-Srivastava
l
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 DziokSrivastava 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]).
2. M AIN R ESULTS
l
In this section we prove some results on the linear operator Hm
[α1 ]. First is the inclusion
theorem.
Theorem 2.1. Let <α1 >
U ST (k, γ).
1−γ
,
k+1
l
l
and f ∈ A. If Hm
[α1 + 1]f ∈ U ST (k, γ) then Hm
[α1 ]f ∈
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 P
be univalently convex in the unit disk U with
n
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
(2.1)
α1
l
l
Hm
[α1 + 1]f (z)
z(Hm
[α1 ]f (z))0
=
+ (α1 − 1) = p(z) + (α1 − 1).
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
From this and the argument given in Section 1 we may write
p(z) +
J. Inequal. Pure and Appl. Math., 6(2) Art. 52, 2005
zp0 (z)
≺ qk,γ (z).
p(z) + (α1 − 1)
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4
R. AGHALARY AND G H . A ZADI
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 >
U CV (k, γ).
1−γ
,
k+1
l
l
and f ∈ A. If Hm
[α1 + 1]f ∈ U CV (k, γ) then Hm
[α1 ]f ∈
Proof. By virtue of (1.1), (1.2) and Theorem 2.1 we have
l
l
Hm
[α1 + 1]f ∈ U CV (k, γ) ⇔ z Hm
[α1 + 1]f
0
∈ U ST (k, γ)
l
⇔ Hm
[α1 + 1]zf 0 ∈ U ST (k, γ)
l
[α1 ]zf 0 ∈ U ST (k, γ)
⇒ Hm
l
⇔ Hm
[α1 ]f ∈ U CV (k, γ).
and the proof is complete.
We next prove
Theorem 2.3. Let <α1 >
U CC(k, γ, β).
1−γ
,
k+1
l
l
and f ∈ A. If Hm
[α1 + 1]f ∈ U CC(k, γ, β) then Hm
[α1 ]f ∈
To prove the above theorem, we shall need the following lemma which is due to Miller and
Mocanu [10].
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)
l
0
l
0
z(Hm [α1 ]g) (z)
m [α1 ]f ) (z)
Letting h(z) = z(H
we observe that h and H are analytic in
l [α ]g)(z) and H(z) =
l [α ]g(z)
(Hm
Hm
1
1
l
U and h(0) = H(0) = 1. Now, by Theorem 2.1, Hm [α1 ]g ∈ U ST (k, β) and so <H(z) > k+β
.
k+1
Also, note that
(2.4)
l
l
z(Hm
[α1 ]f )0 (z) = (Hm
[α1 ]g(z))h(z).
Differentiating both sides of (2.4) yields
l
l
z(Hm
[α1 ](zf 0 ))0 (z)
z(Hm
[α1 ]g)0 (z)
=
h(z) + zh0 (z) = H(z)h(z) + zh0 (z).
l [α ]g(z)
l [α ]g(z)
Hm
Hm
1
1
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T HE D ZIOK -S RIVASTAVA O PERATOR AND k−U NIFORMLY S TARLIKE F UNCTIONS
5
Now using the identity (1.5) we obtain
(2.5)
l
l
z(Hm
[α1 + 1]f )0 (z)
Hm
[α1 + 1](zf 0 )(z)
=
l [α + 1]g(z)
l [α + 1]g(z)
Hm
Hm
1
1
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 )(z)
l [α ](zf 0 ))0 (z)
Hm
z(Hm
1
1
+
(α
−
1)
1
l [α ]g(z)
l
Hm
H
1
m [α1 ]g(z)
l [α ]g)0 (z)
z(Hm
1
+ (α1 − 1)
l [α ]g(z)
Hm
1
0
H(z)h(z) + zh (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
1
h(z) +
zh0 (z) ≺ qk,γ (z).
H(z) + (α1 − 1)
On letting E = 0 and 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.
<(B(z)) =
Using a similar argument to that in Theorem 2.2 we can prove
Theorem 2.4. Let <α1 >
U QC(k, γ, β).
1−γ
,
k+1
l
l
and f ∈ A. If Hm
[α1 + 1]f ∈ U QC(k, γ, β), then Hm
[α1 ]f ∈
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
Lc (f ) =
t f (t)dt, c > −1.
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)
Substituting
l
l
l
z(Hm
[α1 ]Lc (f ))0 (z) = (c + 1)Hm
[α1 ]f (z) − c(Hm
[α1 ]Lc (f ))(z).
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)
(2.7)
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.
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 ).
J. Inequal. Pure and Appl. Math., 6(2) Art. 52, 2005
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6
R. AGHALARY AND G H . A ZADI
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)
≺ qk,γ (z)
l [α ]g)(z)
(Hm
1
(2.8)
(z ∈ U ).
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
l
0
l [α ]L (zf 0 ))0 (z)
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
(2.9)
.
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 defined by
m
l
l
z(Hm
[α1 ]Lc (f ))0 = h(z)Hm
[α1 ]Lc (g).
(2.10)
Differentiating both sides of (2.10) yields
(2.11)
l
l
z(Hm
[α1 ]Lc (g))0 (z)
z(Hm
[α1 ](zLc (f ))0 )0 (z)
0
=
zh
(z)
+
h(z)
= zh0 (z) + H(z)h(z).
l [α ]L (g))(z)
l [α ]L (g))(z)
(Hm
(H
1 c
m 1 c
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
h(z) +
(2.12)
zh0 (z)
≺ qk,γ (z).
H(z) + c
1
Letting B(z) = H(z)+c
in (2.12) we note that <(B(z)) > 0 if c > − k+β
. Now for E = 0 and B
k+1
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
l
l
If Hm
[α1 ]f ∈ U QC(k, γ, β) so is Lc (Hm
[α1 ]f ).
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