Volume 10 (2009), Issue 4, Article 100, 5 pp.
SOME APPLICATIONS OF THE GENERALIZED
BERNARDI–LIBERA–LIVINGSTON INTEGRAL OPERATOR ON UNIVALENT
FUNCTIONS
M. ESHAGHI GORDJI, D. ALIMOHAMMADI, AND A. EBADIAN
D EPARTMENT OF M ATHEMATICS
FACULTY OF S CIENCE
S EMNAN U NIVERSITY
S EMNAN , I RAN
madjideshaghi@gmail.com
D EPARTMENT OF M ATHEMATICS
A RAK U NIVERSITY
A RAK , I RAN
d-alimohammadi@araku.ac.ir
D EPARTMENT OF M ATHEMATICS
FACULTY OF S CIENCE
U RMIA U NIVERSITY
U RMIA , I RAN
ebadian.ali@gmail.com
Received 17 October, 2008; accepted 24 July, 2009
Communicated by S.S. Dragomir
A BSTRACT. In this paper by making use of the generalized Bernardi–Libera–Livingston integral
operator we introduce and study some new subclasses of univalent functions. Also we investigate
the relations between those classes and the classes which are studied by Jin–Lin Liu.
Key words and phrases: Analytic function, Integral operator, Univalent function.
2000 Mathematics Subject Classification. 30C45, 30C50.
1. I NTRODUCTION
P
n
Let A be the class of functions of the form, f (z) = z + ∞
n=2 an z which are analytic in the
unit disk U = {z : |z| < 1}. Also, let S denote the subclass of A consisting of all univalent
functions in U . Suppose λ is a realnnumber
o with 0 ≤ λ < 1. A function f ∈ S is said to be
zf 0 (z)
starlike of order λ if and only if Re f (z) > λ, z ∈ U . Also, f ∈ S is said to be convex of
o
n
zf 00 (z)
order λ if and only if Re 1 + f 0 (z) > λ, z ∈ U . We denote by S ∗ (λ), C(λ) the classes of
starlike and convex functions of order λ respectively. It is well known that f ∈ C(λ) if and only
The authors would like to thank the referee for a number of valuable suggestions regarding of a previous version of this paper.
284-08
2
M. E SHAGHI G ORDJI , D. A LIMOHAMMADI , AND A. E BADIAN
if zf 0∗ (λ).
fo∈ A, then f ∈ K(β, λ) if and only if there exists a function g ∈ S ∗ (λ) such
n If
0
(z)
> β, z ∈ U, where 0 ≤ β < 1. These functions are called close-to-convex
that Re zfg(z)
functions of order β type λ. A function n
f ∈ A isocalled quasi-convex of order β type λ if there
0 (z))0
> β. We denote this class by K ∗ (β, λ) [10].
exists a function g ∈ C(λ) such that Re (zfg0 (z)
It is easy to see that f ∈ K ∗ (β, γ) if and only if zf 0 ∈ K(β, γ) [9]. For f ∈ A if for some
λ(0 ≤ λ < 1) and η(0 < η ≤ 1) we have
0
π
zf (z)
< η, (z ∈ U ),
(1.1)
arg
−
λ
2
f (z)
then f (z) is said to be strongly starlike of order η and type λ in U and we denote this class by
S ∗ (η, λ). If f ∈ A satisfies the condition
π
zf 00 (z)
< η, (z ∈ U )
−
λ
(1.2)
arg
1
+
2
f 0 (z)
for some λ and η as above, then we say that f (z) is strongly convex of order η and type λ in
U and we denote this class by C(η, λ). Clearly f ∈ C(η, λ) if and only if zf 0∗ (η, λ), and in
particular, we have S ∗ (1, λ) = S ∗ (λ) and C(1, λ) = C(λ).
For c > −1 and f ∈ A the generalized Bernardi–Libera–Livingston integral operator Lc f is
defined as follows
Z
c + 1 z c−1
(1.3)
Lc f (z) =
t f (t)dt.
zc 0
This operator for c ∈ N = {1, 2, 3, . . . } was studied by Bernardi [1] and for c = 1 by Libera
[4] (see also [8]). The classes STc (η, λ) and CVc (η, λ) were introduced by Liu [7], where
z(Lc f (z))0
∗
STc (η, λ) = f ∈ A : Lc f ∈ S (η, λ),
6= λ, z ∈ U ,
Lc f (z)
(z(Lc f (z))0 )0
CVc (η, λ) = f ∈ A : Lc f ∈ C(η, λ),
6= λ, z ∈ U .
(Lc f (z))0
Now by making use of the operator given by (1.3) we introduce the following classes.
Sc∗ (λ) = {f ∈ A : Lc f ∈ S ∗ (λ)},
Cc (λ) = {f ∈ A : Lc f ∈ C(λ)}.
Obviously f ∈ CVc (η, λ) if and only if zf 0 ∈ STc (η, λ). J. L. Liu [5] and [6] introduced and
similarly investigated the classes Sσ∗ (λ), Cσ (λ), Kσ (β, λ), Kσ∗ (β, λ), STσ (η, λ), CVσ (η, λ) by
making use of the integral operator I σ f given by
Z z
2σ
z σ−1
σ
(1.4)
I f (z) =
log
f (t)dt, σ > 0, f ∈ A.
zΓ(σ) 0
t
The operator I σ was introduced by Jung, Kim and Srivastava [2] and then investigated by Uralegaddi and Somanatha [13], Li [3] and Liu [5]. For the integral operators given by (1.3) and (1.4)
we have verified following relationships.
σ
∞ X
2
σ
(1.5)
I f (z) = z +
an z n ,
n
+
1
n=2
(1.6)
∞
X
c+1
Lc f (z) = z +
an z n ,
n
+
c
n=2
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 100, 5 pp.
http://jipam.vu.edu.au/
B ERNARDI -L IBERA -L IVINGSTON I NTEGRAL O PERATOR
(1.7)
z(I σ Lc f (z))0σ f (z) − cI σ Lc f (z),
(1.8)
z(Lc I σ f (z))0σ f (z) − cLc I σ f (z).
3
It follows from (1.5) that one can define the operator I σ for any real number σ. In this paper
we investigate the properties of the classes Sc∗ (λ), Cc (λ), Kc (β, λ), Kc∗ (β, λ), STc (η, λ) and
CVc (η, λ). We also study the relations between these classes by the classes which are introduced
by Liu in [5] and [6]. For our purposes we need the following lemmas.
Lemma 1.1 ([9]). Let u = u1 + iu2 , v = v1 + iv2 and let ψ(u, v) be a complex function
ψ : D ⊂ C × C → C. Suppose that ψ satisfies the following conditions
(i) ψ(u, v) is continuous in D;
(ii) (1, 0) ∈ D and Re{ψ(1, 0)} > 0;
1+u2
(iii) Re{ψ(iu2 , v1 )} P
≤ 0 for all (iu2 , v1 ) ∈ D with v1 ≤ − 2 2 .
n
0
Let p(z) = 1 + ∞
n=2 cn z be analytic in U so that (p(z), zp (z)) ∈ D for all z ∈ U . If
0
Re{ψ(p(z), zp (z))} > 0, z ∈ U then Re{p(z)} > 0, z ∈ U.
P
n
Lemma 1.2 ([11]). Let the function p(z) = 1 + ∞
n=1 cn z be analytic in U and p(z) 6= 0,
z ∈ U. If there exists a point z0 ∈ U such that | arg(p(z))| < π2 η for |z| < |z0 | and arg p(z0 )| =
p0 (z0 )
π
η where 0 < η ≤ 1, then z0p(z
= ikη and k ≥ 12 (r + 1r ) when arg p(z0 ) = π2 η, Also,
2
0)
k ≤ −1
(r + 1r ) when arg p(z0 ) = −π
η, and p(z0 )1/η = ±ir(r > 0).
2
2
2. M AIN R ESULTS
In this section we obtain some inclusion theorems by following the method of proof adopted
in [12].
Theorem 2.1.
n
(i) For f ∈ A if Re
∗
Sc∗ (λ) ⊂ Sc+1
(λ).
zf 0 (z)
f (z)
−
z(Lc f (z))0
Lc f (z)
o
> 0 and
n 0
(z)
−
(ii) Let c > −λ. For f ∈ A if Re zff (z)
∗
∗
function, then Sc+1 (λ) ⊂ Sc (λ).
z(Lc+1 f (z))0
Lc+1 f (z)
z(Lc+1 f (z))0
Lc+1 f (z)
o
is an analytic function, then
> 0 and
z(Lc+1 f (z))0
Lc+1 f (z)
is an analytic
Proof. (i) Suppose that f ∈ Sc∗ (λ) and set
z(Lc+1 f (z))0
− λ = (1 − λ)p(z),
Lc+1 f (z)
(2.1)
where p(z) = 1 +
P∞
(2.2)
n=2 cn z
n
. An easy calculation shows that
h
i
z(Lc+1 f (z))0
z(Lc+1 f (z))00
2 + c + (Lc+1 f (z))0
Lc+1 f (z)
zf 0 (z)
=
.
0
z(Lc+1 f (z))
f (z)
+c+1
Lc+1 f (z)
By setting H(z) =
(2.3)
z(Lc+1 f (z))0
Lc+1 f (z)
we have
1+
z(Lc+1 f (z))00
zH 0 (z)
=
H(z)
+
.
(Lc+1 f (z))0
H(z)
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 100, 5 pp.
http://jipam.vu.edu.au/
4
M. E SHAGHI G ORDJI , D. A LIMOHAMMADI , AND A. E BADIAN
By making use of (2.3) in (2.2) , since H(z) = λ + (1 − λ)p(z), we obtain
(1 − λ)p(z) +
(2.4)
(1 − λ)zp0 (z)
zf 0 (z)
=
− λ.
λ + c + 1 + (1 − λ)p(z)
f (z)
If we consider
(1 − λ)v
ψ(u, v) = (1 − λ)u +
,
λ + c + 1 + (1 − λ)u
×C and (1, 0) ∈ D. Also, ψ(1, 0) >
then ψ(u, v) is a continuous function in D = C − λ+c+1
λ−1
0 and for all (iu2 , v1 ) ∈ D with v1 ≤ −
1+u22
2
we have
(1 − λ)(λ + c + 1)v1
(1 − λ)2 u22 + (λ + c + 1)2
−(1 − λ)(λ + c + 1)(1 + u22 )
≤
< 0.
2[(1 − λ)2 u22 + (λ + c + 1)2 ]
Re ψ(iu2 , v1 ) =
Therefore the function ψ(u, v) satisfies the conditions of Lemma 1.1 and since in view of the
assumption, by considering (2.4) , we have Re{ψ(p(z), zp0 (z))} > 0, Lemma 1.1 implies that
Re p(z) > 0, z ∈ U and this completes the proof of (i).
(ii) For proving this part of the theorem, we use the same method and a easily verified formula
similar to (2.2). By replacing c + 1 with c we get the desired result.
Theorem 2.2.
n
zf 0 (z)
f (z)
z(Lc f (z))0
Lc f (z)
o
0
c+1 f (z))
(i) For f ∈ A if Re
−
> 0 and z(L
is an analytic function, then
Lc+1 f (z)
Cc (λ) ⊂ Cc+1 (λ).
n 0
o
0
0
(z)
c+1 f (z))
c+1 f (z))
(ii) Let c > −λ. For f ∈ A if Re zff (z)
− z(L
> 0 and z(L
is an analytic
Lc+1 f (z)
Lc+1 f (z)
function, then Cc+1 (λ) ⊂ Cc (λ).
Proof. (i) In view of part (i) of Theorem 2.1 we can write
0∗
f ∈ Cc (λ) ⇔ Lc f ∈ C(λ) ⇔ z(Lc f )0∗ (λ) ⇔ Lc zf 0∗ (λ) ⇔ zfc0∗ (λ) ⇒ zfc+1
(λ)
⇔ Lc+1 zf 0∗ (λ) ⇔ z(Lc+1 f )0∗ (λ) ⇔ Lc+1 f ∈ C(λ) ⇔ f ∈ Cc+1 (λ).
Part (ii) of the theorem can be proved in a similar manner.
Theorem 2.3. If c ≥ −λ and
zf 0 (z)
f (z)
is an analytic function, then f ∈ S ∗ (λ) implies f ∈ Sc∗ (λ).
Proof. By differentiating logarithmically both sides of (1.3) with respect to z we obtain
z(Lc f (z))0
(c + 1)f (z)
+c=
.
Lc f (z)
Lc f (z)
(2.5)
Again differentiating logarithmically both sides of (2.5) we have
p(z) +
(2.6)
zp0 (z)
zf 0 (z)
=
− λ,
c + λ + p(z)
f (z)
0
v
c f (z))
where p(z) = z(L
− λ. Let us consider ψ(u, v) = u + u+c+λ
. Then ψ is a continuous
Lc f (z)
function in D = {C − (−c − λ)} × C, (1, 0) ∈ D and Re ψ(1, 0) > 0. If (iu2 , v1 ) ∈ D with
1+u2
v1 ≤ − 2 2 , then
v1 (c + λ)
Re ψ(iu2 , v1 ) = 2
≤ 0.
u2 + (c + λ)2
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 100, 5 pp.
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B ERNARDI -L IBERA -L IVINGSTON I NTEGRAL O PERATOR
5
Since f ∈ S ∗ (λ), then (2.6) gives
0
Re(ψ(p(z), zp (z))) = Re
zf 0 (z)
−λ
f (z)
> 0.
Therefore Lemma 1.1 concludes that Re{p(z)} > 0 and this completes the proof.
Corollary 2.4. If c ≥ λ and
zf 0 (z)
f (z)
is an analytic function, then f ∈ C(λ) implies f ∈ Cc (λ).
Proof. We have
f ∈ C(λ) ⇔ zf 0∗ (λ)Λzfc0∗ (λ) ⇔ Lc zf 0 ∈ S ∗ (λ)
⇔ z(Lc f )0∗ (λ) ⇔ Lc f ∈ C(λ) ⇔ f ∈ Cc (λ).
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J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 100, 5 pp.
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