Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 3 Issue 4(2011), Pages 155-162.
INCLUSION PROPERTIES FOR CERTAIN K-UNIFORMLY
SUBCLASSES OF ANALYTIC FUNCTIONS ASSOCIATED WITH
CERTAIN INTEGRAL OPERATOR
(COMMUNICATED BY JOSZEF SANDOR)
M. K. AOUF AND T. M. SEOUDY
Abstract. In this paper, we introduce several new k-uniformly classes of analytic functions defined by using the integral operator and investigate various
inclusion relationships for these classes. Some interesting applications involving certain classes of integral operators are also considered.
1. Introduction
Let A denote the class of functions of the form
∞
∑
f (z) = z +
an z n
(1.1)
n=2
which are analytic in the open unit disk U = {z ∈ C : |z| < 1}. If f and g are
analytic in U, we say that f is subordinate to g, written f ≺ g or f (z) ≺ g(z),
if there exists a Schwarz function ω, analytic in U with ω (0) = 0 and |ω (z)| < 1
(z ∈ U), such that f (z) = g(ω(z)) (z ∈ U). In particular, if the function g is
univalent in U, the above subordination is equivalent to f (0) = g(0) and f (U) ⊂
g(U) (see [9] and [10]). For 0 ≤ γ, β < 1, we denote by S ∗ (γ) , C (γ) , K (γ, β)
and K ∗ (γ, β) the subclasses of A consisting of all analytic functions which are,
respectively, starlike of order γ, convex of order γ, close-to-convex of order γ, and
type β and quasi-convex of order γ, and type β in U.
Now, we introduce the subclasses U S ∗ (k; γ), U C (k; γ), U K (k; γ, β) and U K ∗ (k; γ, β)
of the class A for 0 ≤ γ, β < 1, and k ≥ 0, which are defined by
′
}
{
( ′
)
zf (z)
zf (z)
∗
U S (k; γ) = f ∈ A : ℜ
−γ > k
− 1 ,
(1.2)
f (z)
f (z)
2000 Mathematics Subject Classification. 30C45.
Key words and phrases. Analytic functions, k-uniformly starlike functions, k-uniformly convex functions, k-uniformly close-to-convex functions, k-uniformly quasi-convex functions, integral
operator, Hadamard product, subordination.
c
⃝2011
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted December 7, 2009. Published 26 October 2011.
155
156
M. K. AOUF AND T. M. SEOUDY
′′
}
zf (z) U C (k; γ) =
> k ′
(1.3)
+γ ,
f (z) ′
}
{
( ′
)
zf (z)
zf
(z)
U K (k; γ, β) = f ∈ A : ∃ g ∈ U S ∗ (k; β) s.t. ℜ
−γ > k
− 1 ,
g (z)
g (z)
(1.4)

(


(
)′
)′

′
′




zf (z)
 zf (z)
∗
U K (k; γ, β) = f ∈ A : ∃ g ∈ U C (k; γ) s.t.ℜ 
−
γ
−
1
>
k
.

′
′


g
(z)
g
(z)


(1.5)
We note that
U S ∗ (0; γ) = S ∗ (k; γ) , U C (0; γ) = C (γ) ,
U K (0; γ, β) = K (γ, β) , U K ∗ (0; γ, β) = K ∗ (γ, β) (0 ≤ γ, β < 1) .
{
(
′′
zf (z)
f ∈A:ℜ 1+ ′
−γ
f (z)
)
Corresponding to a conic domain Ωk,γ defined by
{
}
√
2
Ωk,γ = u + iv : u > k (u − 1) + v 2 + γ ,
(1.6)
we define the function qk,γ (z) which maps U onto the conic domain Ωk,γ such that
1 ∈ Ωk,γ as the following:
 1+(1−2γ)z
(k = 0) ,

1−z {

√ }

( −1 )

1+√z
1−γ
k2 −γ
2

(0 < k < 1) ,

 1−k2 cos π cos k i log 1− z − 1−k2
(
√ )2
qk,γ (z) =
(1.7)
2(1−γ)
1+√z
1 + π2
log 1− z
(k = 1) ,


{
}


√
∫ u(z)

2

π
dt
k
 1−γ
√
sin
+ kk2−γ
(k > 1) .
√
2
k −1
2ζ(k) 0
−1
2
2 2
1−t
1−k t
√
′
z− k
πζ (z)
√
where u (z) =
and ζ (k) is such that k = cosh
. By virture of the
4ζ (z)
1 − kz
properties of the conic domain Ωk,γ , we have
k+γ
.
(1.8)
k+1
Making use of the principal of subordination and the definition of qk,γ (z), we
may rewrite the subclasses U S ∗ (k; γ), U C (k; γ), U K (k; γ, β) and U K ∗ (k; γ, β) as
the following:
{
}
′
zf (z)
∗
(1.9)
U S (k; γ) = f ∈ A :
≺ qk,γ (z) ,
f (z)
}
{
′′
zf (z)
≺ qk,γ (z) ,
(1.10)
U C (k; γ) = f ∈ A : 1 + ′
f (z)
{
}
′
zf (z)
∗
U K (k; γ, β) = f ∈ A : ∃ g ∈ U S (k; β) s.t.
≺ qk,γ (z) ,
(1.11)
g (z)


( ′ )′




zf (z)
U K ∗ (k; γ, β) = f ∈ A : ∃ g ∈ U C (k; γ) s.t.
≺
q
(z)
.
(1.12)
k,γ


g ′ (z)


ℜ {qk,γ (z)} >
INCLUSION PROPERTIES FOR CERTAIN K-UNIFORMLY...
157
Recently, Komatu [5] introduced a certain integral operator Lλa : A → A (a > 0; λ ≥ 0)
as follows:
L0a f (z) = f (z) (a > 0; λ = 0)
(1.13)
and
(
)λ−1
λ ∫ 1
(1 + a)
1
ta−1 log
Lλa f (z) =
f (tz) dt (a > 0; λ > 0) .
(1.14)
Γ (λ)
t
0
Thus, if f ∈ A is of the form (1.1), it is easily seen from (1.13) and (1.14) that
)λ
∞ (
∑
a+1
Lλa f (z) = z +
an z n (a > 0; λ ≥ 0) .
(1.15)
a
+
n
n=2
It is easily to deduce from (1.15) that
(
)′
z Lλ+1
f (z) = (a + 1) Lλa f (z) − aLλ+1
f (z) .
a
a
(1.16)
Lλa
The special case a = 1 of the inegral operator
is essentially the operator which
considered by Jung et al. [4].
Next, by using the operator Lλa , we introduce the following classes of analytic
functions for a > 0, λ ≥ 0, k ≥ 0 and 0 ≤ γ, β < 1:
}
{
(1.17)
U S ∗ (λ; k; γ) = f ∈ A : Lλa f (z) ∈ U S ∗ (k; γ) ,
}
{
(1.18)
U C (λ; k; γ) = f ∈ A : Lλa f (z) ∈ U C (k; γ) ,
}
{
λ
(1.19)
U K (λ; k; γ, β) = f ∈ A : La f (z) ∈ U K (k; γ, β) ,
}
{
∗
∗
λ
(1.20)
U K (λ; k; γ, β) = f ∈ A : La f (z) ∈ U K (k; γ, β) .
We also note that
′
f (z) ∈ U S ∗ (λ; k; γ) ⇔ zf (z) ∈ U C (λ; k; γ) ,
and
(1.21)
′
f (z) ∈ U K (λ; k; γ, β) ⇔ zf (z) ∈ U K ∗ (λ; k; γ, β) .
(1.22)
∗
In this paper, we investgate several inclusion properties of the classes U S (λ; k; γ),
U C (λ; k; γ), U K (λ; k; γ, β) and U K ∗ (λ; k; γ, β) associated with the operator Lλa .
Some applications involving integral operators are also considered.
2. Inclusion Properties Involving the Operator Lλa
In order to prove the main results, we shall need The following lemmas.
Lemma 1[3]. Let h (z) be convex univalent in U with h (0) = 1 and ℜ {ηh (z) + γ} >
0 (η, γ ∈ C). If p (z) is analytic in U with p (0) = 1, then
′
p (z) +
zp (z)
≺ h (z)
ηp (z) + γ
(2.1)
implies
p (z) ≺ h (z) .
(2.2)
Lemma 2[8]. Let h (z) be convex univalent in U and let w be analytic in U
with ℜ {w (z)} ≥ 0. If p (z) is analytic in U and p (0) = h (0), then
′
p (z) + w (z) zp (z) ≺ h (z)
(2.3)
p (z) ≺ h (z) .
(2.4)
implies
158
M. K. AOUF AND T. M. SEOUDY
Theorem 1. U S ∗ (λ; k; γ) ⊂ U S ∗ (λ + 1; k; γ) .
Proof. Let f ∈ U S ∗ (λ; k; γ) and set
(
)′
z Lλ+1
f (z)
a
p (z) =
Lλ+1
f (z)
a
(z ∈ U) ,
(2.5)
where p (z) is analytic in U with p (0) = 1. From (1.16) and (2.5), we have
Lλa f (z)
1
{p (z) + a} .
=
λ+1
a
+
1
La f (z)
(2.6)
Differentiating (2.6) with respect to z and multiplying the result equation by z, we
obtain
(
)′
′
z Lλa f (z)
zp (z)
=
p
(z)
+
.
(2.7)
Lλa f (z)
p (z) + a
From this and the argument given in Section 1, we may write
′
p (z) +
zp (z)
≺ qk,γ (z)
p (z) + a
(z ∈ U) .
(2.8)
k+γ
, we see that
k+1
Since a > 0 and ℜ {qk,γ (z)} >
ℜ {qk,γ (z) + a} > 0
(z ∈ U) .
(2.9)
Applying Lemma 1 to (2.8), it follows that p (z) ≺ qk,γ (z), that is, f ∈ U S ∗ (λ + 1; k; γ).
Theorem 2. U C (λ; k; γ) ⊂ U C (λ + 1; k; γ) .
Proof. Applying (1.21) and Theorem 1, we observe that
f (z)
∈
=⇒
′
U C (λ; k; γ) ⇐⇒ zf (z) ∈ U S ∗ (λ; k; γ)
′
zf (z) ∈ U S ∗ (λ + 1; k; γ)
⇐⇒ f (z) ∈ U C (λ + 1; k; γ) ,
which evidently proves Theorem 2.
Theorem 3. U K (λ; k; γ, β) ⊂ U K (λ + 1; k; γ, β) .
Proof. Let f ∈ U K (λ; k; γ, β). Then, from the definition of U K (λ; k; γ, β), there
exists a function r (z) ∈ U S ∗ (k; γ) such that
(
)′
z Lλa f (z)
≺ qk,γ (z) .
(2.10)
r (z)
Choose the function g (z) such that Lλa g (z) = r (z). Then, g ∈ U S ∗ (λ; k; γ) and
(
)′
z Lλa f (z)
≺ qk,γ (z) .
(2.11)
Lλa g (z)
Now let
p (z) =
(
)′
z Lλ+1
f (z)
a
Lλ+1
g (z)
a
,
(2.12)
INCLUSION PROPERTIES FOR CERTAIN K-UNIFORMLY...
159
where p (z) is analytic in U with p (0) = 1. Since g ∈ U S ∗ (λ; k; γ), by Theorem 1,
we know that g ∈ U S ∗ (λ + 1; k; γ). Let
(
)′
z Lλ+1
g (z)
a
t (z) =
(z ∈ U) ,
(2.13)
Lλ+1
g (z)
a
where t (z) is analytic in U with ℜ {t (z)} > k+γ
k+1 . Also, from (2.13), we note that
(
)′
(
)
′
z Lλ+1
f (z) = Lλ+1
zf (z) = Lλ+1
g (z) p (z) .
(2.14)
a
a
a
Differentiating both sides of (2.14) with respect to z and multiplying the result
equation by z, we obtain
(
)′
′
(
)′
z Lλ+1
zf
(z)
a
z Lλ+1
g (z)
′
′
a
=
p (z) + zp (z) = t (z) p (z) + zp (z) . (2.15)
λ+1
λ+1
La g (z)
La g (z)
Now using the identity (1.16) and (2.15), we obtain
(
)′
′
′
(
)′
λ+1
′
λ
z
L
zf
(z)
+ aLλ+1
zf (z)
a
a
z Lλa f (z)
La zf (z)
=
=
(
)′
Lλa g (z)
Lλa g (z)
z Lλ+1
g (z) + aLλ+1
g (z)
a
a
(
)′
′
z Lλ+1
zf (z)
a
g(z)
Lλ+1
a
=
′
+a
g(z)
Lλ+1
a
g(z)
Lλ+1
a
′
z(
z (Lλ+1
f (z))
a
g(z)
Lλ+1
a
′
)
+a
=
t (z) p (z) + zp (z) + ap (z)
t (z) + a
=
p (z) +
′
zp (z)
.
t (z) + a
k+γ
, we see that
k+1
ℜ {t (z) + a} > 0 (z ∈ U) .
(2.16)
Since a > 0 and ℜ {t (z)} >
(2.17)
Hence, applying Lemma 2, we can show that p (z) ≺ qk,γ (z) so that f ∈ U K (λ + 1; k; γ, β).
This completes the proof of Theorem 3.
Theorem 4. U K ∗ (λ; k; γ, β) ⊂ U K ∗ (λ + 1; k; γ, β) .
Proof. Just as we derived Theorem 2 as consequence of Theorem 1 by using the
equivalence (1.21), we can also prove Theorem 4 by using Theorem 3 and the
equivalence (1.22).
3. Inclusion Properties Involving the Integral Operator Fc
In this section, we consider the generalized Libera integral operator Fc ( see [2], [6] and [7])
defined by
∫
c + 1 z c−1
t f (t) dt
(f ∈ A; c > −1) .
(3.1)
Fc (f ) (z) =
zc
0
∗
∗
Theorem 5. Let c > − k+γ
k+1 . If f ∈ U S (λ; k; γ), then Fc (f ) ∈ U S (λ; k; γ) .
160
M. K. AOUF AND T. M. SEOUDY
Proof. Let f ∈ U S ∗ (λ; k; γ) and set
(
)′
z Lλa Fc (f ) (z)
p (z) =
Lλa Fc (f ) (z)
(z ∈ U) ,
where p (z) is analytic in U with p (0) = 1. From (3.1), we have
(
)′
z Lλa Fc (f ) (z) = (c + 1) Lλa f (z) − cLλa Fc (f ) (z) .
(3.2)
(3.3)
Then, by using (3.2) and (3.3), we obtain
(c + 1)
Lλa f (z)
λ
La Fc (f ) (z)
= p (z) + c.
(3.4)
Taking the logarithmic differentiation on both sides of (3.4) and multiplying by z,
we have
(
)′
′
z Lλa f (z)
zp (z)
p (z) +
=
≺ qk,γ (z) .
(3.5)
p (z) + c
Lλa f (z)
Hence, by virtue of Lemma 1, we conclude that p (z) ≺ qk,γ (z) in U, which implies
that Fc (f ) ∈ U S ∗ (λ; k; γ).
Theorem 6. Let c > − k+γ
k+1 . If f ∈ U C (λ; k; γ), then Fc (f ) ∈ U C (λ; k; γ).
Proof. By applying Theorem 5, it follows that
f (z)
∈
=⇒
′
U C (λ; k; γ) ⇐⇒ zf (z) ∈ U S ∗ (λ; k; γ)
( ′)
Fc zf (z) ∈ U S ∗ (λ; k; γ) (by T heorem 5)
′
⇐⇒ z (Fc (f ) (z)) ∈ U S ∗ (λ; k; γ)
(3.6)
⇐⇒ Fc (f ) (z) ∈ U C (λ; k; γ) ,
which proves Theorem 6.
Theorem 7. Let c > − k+γ
k+1 . If f ∈ U K (λ; k; γ, β), then Fc (f ) ∈ U K (λ; k; γ, β).
Proof. Let f ∈ U K (λ; k; γ, β). Then, in view of the definition of the class U K (λ; k; γ, β),
there exists a function g ∈ U S ∗ (λ; k; γ) such that
)′
(
z Lλa f (z)
≺ qk,γ (z) .
(3.7)
Lλa g (z)
Thus, we set
(
)′
z Lλa Fc (f ) (z)
p (z) =
(z ∈ U) ,
(3.8)
Lλa Fc (g) (z)
where p (z) is analytic in U with p (0) = 1. Since g ∈ U S ∗ (λ; k; γ), we see from
Theorem 5 that Fc (g) ∈ U S ∗ (λ; k; γ). Using (3.3) and let
(
)′
z Lλa Fc (g) (z)
,
(3.9)
t (z) =
Lλa Fc (g) (z)
k+γ
. Using (3.8), we have
k+1
(
)
′
Lλa zFc (f ) (z) = Lλa Fc (g) (z) p (z) .
where t (z) is analytic in U with ℜ {t (z)} >
(3.10)
INCLUSION PROPERTIES FOR CERTAIN K-UNIFORMLY...
161
Differentiating both sides of (3.10) with respect to z and multiplying by z, we obtain
(
)′
′
(
)′
z Lλa zFc (f ) (z)
z Lλa Fc (f ) (z)
′
=
p (z) + zp (z)
Lλa Fc (g) (z)
Lλa Fc (g) (z)
′
= t (z) p (z) + zp (z) .
(3.11)
Now using the identity (3.3) and (3.11), we obtain
(
)′
′
′
( λ
)′
λ
′
z
L
zF
(f
)
(z)
+ cLλa zFc (f ) (z)
λ
a
c
z La f (z)
La zf (z)
=
=
′
Lλa g (z)
Lλa g (z)
z (Lλa Fc (g) (z)) + cLλa Fc (g) (z)
)′
(
′
(
)′
z Lλa zFc (f ) (z)
z Lλa Fc (f ) (z)
+c
Lλa Fc (g) (z)
Lλa Fc (g) (z)
=
( λ
)′
z La Fc (g) (z)
+c
Lλa Fc (g) (z)
′
=
t (z) p (z) + zp (z) + cp (z)
t (z) + c
′
zp (z)
= p (z) +
.
t (z) + c
Since c > −
k+γ
k+γ
and ℜ {t (z)} >
, we see that
k+1
k+1
ℜ {t (z) + c} > 0 (z ∈ U) .
(3.12)
(3.13)
Applying Lemma 2 to (3.12), it follows that p (z) ≺ qk,γ (z) , that is Fc (f ) ∈
U K (λ; k; γ, β).
∗
∗
Theorem 8. Let c > − k+γ
k+1 . If f ∈ U K (λ; k; γ, β), then Fc (f ) ∈ U K (λ; k; γ, β).
Proof. Just as we derived Theorem 6 as consequence of Theorem 5 and (1.21),
we easily deduce the integral-preserving property asserted by Theorem 8 by using
Theorem 7 and (1.22).
Remark. Putting a = 1 in the above results,we obtain the results of Aghalary
and Jahangiri [1].
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M. K. AOUF AND T. M. SEOUDY
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M. K. Aouf, Department of Mathematics, Faculty of Science,, Mansoura University,
Mansoura 35516, Egypt
E-mail address: mkaouf127@yahoo.com
T. M. Seoudy, Department of Mathematics, Faculty of Science,, Fayoum University,
Fayoum 63514, Egypt
E-mail address: tmseoudy@gmail.com, tms00@fayoum.edu.eg