art200805. 114KB Jun 04 2011 12:08:21 AM

ACTA UNIVERSITATIS APULENSIS
No 11/2006
Proceedings of the International Conference on Theory and Application of
Mathematics and Informatics ICTAMI 2005 - Alba Iulia, Romania

ON A CLASS OF α-CONVEX FUNCTIONS
Mugur Acu
Abstract. In this paper we define a general class of α-convex functions
with respect to a convex domain D contained in the right half plane by using
a generalized S˘al˘agean operator introduced by F.M. Al-Oboudi in [5] and we
give some properties of this class.
2000 Mathematical Subject Classification: 30C45
Key words and phrases: α-convex functions, Libera-Pascu integral operator, Briot-Bouquet differential subordination, generalized S˘al˘agean operator
1.Introduction
Let H(U ) be the set of functions which are regular in the unit disc U ,
A = {f ∈ H(U ) : f (0) = f ′ (0) − 1 = 0}, Hu (U ) = {f ∈ H(U ) : f is univalent
in U } and S = {f ∈ A : f is univalent in U }.
Let Dn be the S˘al˘agean differential operator ([10]) defined as:
Dn : A → A, n ∈ N
and
D0 f (z) = f (z)

D1 f (z) = Df (z) = zf ′ (z)
Dn f (z) = D(Dn−1 f (z)).
Remark 1.1 If f ∈ S, f (z) = z+

∞
P

j=2

z+

∞
P

j=2

j n aj z j .

aj z j , j = 2, 3, ..., z ∈ U then Dn f (z) =

The aim of this paper is to define a general class of α-convex functions
with respect to a convex domain D contained in the right half plane by using
a generalized S˘al˘agean operator introduced by F.M. Al-Oboudi in[5] and to
obtain some leftoperties of this class.
7

M. Acu - On a class of α-convex functions
2.Preliminary results
We recall here the definitions of the well - known classes of starlike functions, convex functions and α-convex functions (see [6])
(

)

zf ′ (z)
S = f ∈ A : Re
> 0, z ∈ U ,
f (z)
∗

(

(

zf ′′ (z)
S = CV = K = f ∈ H(U ); f (0) = f (0) − 1 = 0, Re 1 + ′
f (z)
c

′

)

)

> 0, z ∈ U ,

Mα = {f ∈ H(U ), f (0) = f ′ (0) − 1 = 0, ReJ(α, f ; z) > 0, z ∈ U , α ∈ R}
where
zf ′′ (z)
zf ′ (z)

+α 1+ ′
J (α, f ; z) = (1 − α)
f (z)
f (z)

!

We observe that M0 = S ∗ and M1 = S c where S ∗ and S c are the class of
starlike functions, respectively the class of convex functions.
Remark 2.1. By using the subordination relation, we may define the
class Mα thus if f (z) = z + a2 z 2 + ..., z ∈ U , then f ∈ Mα if and only if
1+z
J(α, f ; z) ≺ 1−z
, z ∈ U , where by ”≺” we denote the subordination relation.
Let consider the Libera-Pascu integral operator Lα : A → A defined as:
z

1+aZ
F (t) · ta−1 dt, α ∈ C , Reα ≥ 0.
f (z) = Lα F (z) =

a
z

(1)

0

In the case a = 1, 2, 3, ... this operator was introduced by S. D. Bernardi
and it was studied by many authors in different general cases.
Definition 2.1.[5] Let n ∈ N and λ ≥ 0. We denote with Dλn the operator
defined by
Dλn : A → A ,
Dλ0 f (z) = f (z), Dλ1 f (z) = (1 − λ)f (z) + λzf ′ (z) = Dλ f (z),




Dλn f (z) = Dλ Dλn−1 f (z) .
8

M. Acu - On a class of α-convex functions
Remark 2.2.[5] We observe that Dλn is a linear operator and for f (z) =
z+

∞
P

j=2

aj z j we have
Dλn f (z) = z +

∞
X

(1 + (j − 1)λ)n aj z j .

j=2

Also, it is easy to observe that if we consider λ = 1 in the above definition

we obtain the S˘al˘agean differential operator.
Definition 2.2 [3] Let q(z) ∈ Hu (U ), with q(0) = 1 and q(U ) = D, where
D is a convex domain contained in the right half plane, n ∈ N and λ ≥ 0. We
Dn+1 f (z)
say that a function f (z) ∈ A is in the class SL∗n (q) if Dλ n f (z) ≺ q(z), z ∈ U .
λ

Remark 2.3. Geometric interpretation: f (z) ∈ SL∗n (q) if and only if

n+1
Dλ
f (z)
n f (z)
Dλ

take all values in the convex domain D contained in the right half-

plane.
Definition 2.3 [4] Let q(z) ∈ Hu (U ), with q(0) = 1 and q(U ) = D, where
D is a convex domain contained in the right half plane, n ∈ N and λ ≥ 0. We

Dn+2 f (z)
say that a function f (z) ∈ A is in the class SLcn (q) if Dλn+1 f (z) ≺ q(z), z ∈ U .
λ

Remark 2.4. Geometric interpretation: f (z) ∈ SLcn (q) if and only if

n+2
Dλ
f (z)
n+1
Dλ f (z)

take all values in the convex domain D contained in the right half-

plane.
The next theorem is result of the so called ”admissible functions method”
introduced by P.T. Mocanu and S.S. Miller (see [7], [8], [9]).
Theorem 2.1. Let h convex in U and Re[βh(z) + γ] > 0, z ∈ U . If
p ∈ H(U ) with p(0) = h(0) and p satisfied the Briot-Bouquet differential
subordination

p(z) +

zp′ (z)
≺ h(z), then p(z) ≺ h(z).
βp(z) + γ

9

M. Acu - On a class of α-convex functions
3. Main results
Definition 3.1 Let q(z) ∈ Hu (U ), with q(0) = 1, q(U ) = D, where D is a
convex domain contained in the right half plane, n ∈ N, λ ≥ 0 and α ∈ [0, 1].
We say that a function f (z) ∈ A is in the class M Ln,α (q) if
Jn,λ (α, f ; z) = (1 − α)

Dλn+1 f (z)
Dλn+2 f (z)
+
α
≺ q(z), z ∈ U.

Dλn f (z)
Dλn+1 f (z)

Remark 3.1 Geometric interpretation: f (z) ∈ M Ln,α (q) if and only if
Jn,λ (α, f ; z) take all values in the convex domain D contained in the right
half-plane.
Remark 3.2 It is easy to observe that if we choose different function q(z)
we obtain variously classes of α-convex functions, such as (for example), for
λ = 1 and n = 0, the class of α-convex functions, the class of α-uniform
convex functions with respect to a convex domain (see [2]), and, for λ = 1,
the class U Dn,α (β, γ), β ≥ 0, γ ∈ [−1, 1), β + γ ≥ 0 (see [1]), the class of
α-n-uniformly convex functions with respect to a convex domain (see [2]).
Remark 3.3 We have M Ln,0 (q) = SL∗n (q) and M Ln,1 (q) = SLcn (q).
Remark 3.4 For q1 (z) ≺ q2 (z) we have M Ln,α (q1 ) ⊂ M Ln,α (q2 ) . From
1+z
the above we obtain M Ln,α (q) ⊂ M Ln,α 1−z
Theorem 3.1 For all α, α′ ∈ [0, 1], with α < α′ , we have M Ln,α′ (q) ⊂
M Ln,α (q).
Proof. From f (z) ∈ M Ln,α′ (q) we have
Dλn+2 f (z)

Dλn+1 f (z)
≺ q(z),
+ α n+1
Jn,λ (α, f ; z) = (1 − α)
Dλn f (z)
Dλ f (z)

(2)

where q(z) is univalent in U with q(0) = 1 and maps the unit disc U into the
convex domain D contained in the right half-plane.
With notation
p(z) =

Dλn+1 f (z)
Dλn f (z)
10

M. Acu - On a class of α-convex functions
where
p(z) = 1 + p1 z + . . . andf (z) = z +

∞
X

aj z j

j=2

we have
p(z) + α′ λ ·

=

Dλn+1 f (z)
Dλn f (z)

=

+

Dλn+1 f (z)
Dλn f (z)

Dλn f (z)
α′ λ n+1
Dλ f (z)

+

·z

Dλn f (z)
′
α λ n+1
Dλ f (z)



′

Dλn+1 f (z) Dλn f (z) − Dλn+1 f (z) (Dλn f (z) ′

 

n+1
 z Dλ f (z)



z+

z

Dλn f (z) 
Dn+1 f (z)


+ α′ λ · n+1
= λn
Dλ f (z)
Dλ f (z) 


∞
P

z z+

j=2



z 1+



′

−

∞
P

Dλn+1 f (z)
Dλn f (z)

·

z

(Dλn f (z) ′ 

Dλn f (z)

n+1

(1 + (j − 1)λ)

j=2

n+1

j (1 + (j − 1)λ)

j=2

Dλn f (z)

n

j (1 + (j − 1)λ) aj z

11

Dλn f (z)

−




=



j=2

∞
P

!′

j−1

!






aj z

j−1

!

=

=

!′ 

Dλn f (z)

∞
P

aj z

j

Dλn f (z)

(1 + (j − 1)λ)n aj z j

1+

z

Dn+1 f (z)
Dλn f (z) 

= λn
+ α′ λ · n+1
Dλ f (z)
Dλ f (z) 

Dn+1 f (z)
·
− λn
Dλ f (z)

(Dλn f (z))2

Dλn f (z)



Dn+1 f (z)
·
− λn
Dλ f (z)

zp′ (z)
=
p(z)

−

M. Acu - On a class of α-convex functions
or

(z)

p(z)+α′ ·λ·

zp
=
p(z)

−

Dλn+1 f (z) ′ Dλn f (z)
+α λ· n+1
Dλn f (z)
Dλ f (z)

Dλn+1 f (z)
Dλn f (z)

z+

∞
P

j=2

z+

n+1

j (1 + (j − 1)λ)

j

aj z = z +



j (1 + (j − 1)λ)n aj z j 
Dλn f (z)

We have
∞
X

∞
P
j (1 + (j − 1)λ)n+1 aj z j
z +
j=2

−


Dλn f (z)



∞
X





(3)

((j − 1) + 1) (1 + (j − 1)λ)n+1 aj z j =

j=2

j=2

=z+

∞
X

n+1

(1 + (j − 1)λ)

j

aj z +

∞
X

(j − 1) (1 + (j − 1)λ)n+1 aj z j =

j=2

j=2

= z + Dλn+1 f (z) − z +

∞
X

(j − 1) (1 + (j − 1)λ)n+1 aj z j =

j=2

= Dλn+1 f (z) +
= Dλn+1 f (z) +
= Dλn+1 f (z) −

∞
1X
((j − 1)λ) (1 + (j − 1)λ)n+1 aj z j =
λ j=2

∞
1X
(1 + (j − 1)λ − 1) (1 + (j − 1)λ)n+1 aj z j =
λ j=2

∞
∞
1X
1X
(1 + (j − 1)λ)n+1 aj z j +
(1 + (j − 1)λ)n+2 aj z j =
λ j=2
λ j=2



1  n+1
1  n+2
Dλ f (z) − z +
Dλ f (z) − z =
λ
λ
z
z
1
1
= Dλn+1 f (z) − Dλn+1 f (z) + + Dλn+2 f (z) − =
λ
λ λ
λ
λ − 1 n+1
1
=
Dλ f (z) + Dλn+2 f (z) =
λ
λ

= Dλn+1 f (z) −

12

M. Acu - On a class of α-convex functions

=
Similarly we have
z+

∞
X


1
λ − 1)Dλn+1 f (z) + Dλn+2 f (z) .
λ

j (1 + (j − 1)λ)n aj z j =

j=2


1
λ − 1)Dλn f (z) + Dλn+1 f (z) .
λ

From (3) we obtain
p(z) + α′ · λ ·
=

zp′ (z)
=
p(z)

Dλn+1 f (z)
Dλn f (z) 1
Dλn+1 f (z)
′
+
α
λ
·
(λ
−
1)
+
Dλn f (z)
Dλn f (z)
Dλn+1 f (z) λ

Dλn+2 f (z)
+ n
Dλ f (z)

−

Dλn+1 f (z)
(λ
Dλn f (z)

− 1) −

Dλn+1 f (z)
Dλn f (z)

!2 
=

n+2
n+1
Dλn+1 f (z)
′ Dλ f (z)
′ Dλ f (z)
+ α n+1
=
−α
=
Dλn f (z)
Dλn f (z)
Dλ f (z)

=

n+2
Dλn+1 f (z)
′
′ Dλ f (z)
(1
−
α
)
+
α
= Jn,λ (α′ , f ; z)
Dλn f (z)
Dλn+1 f (z)

From (2) we have
p(z) +

zp′ (z)
≺ q(z)
1
· p(z)
α′ λ

with p(0) = q(0), Req(z) > 0, z ∈ U , α′ > 0 and λ ≥ 0. In this conditions
from Theorem 2.1 we obtain p(z) ≺ q(z) or p(z) take all values in D.
If we consider the function g : [0, α′ ] → C,
g(u) = p(z) + u ·

λzp′ (z)
,
p(z)

with g(0) = p(z) ∈ D and g(α′ ) = Jn,λ (α′ , f ; z) ∈ D, it easy to see that
λzp′ (z)
g(α) = p(z) + α ·
∈ D, 0 ≤ α < α′ .
p(z)
Thus we have
Jn,λ (α, f ; z) ≺ q(z)
13

M. Acu - On a class of α-convex functions
or
f (z) ∈ M Ln,α (q).
From the above theorem we have
Corollary 3.1 For every n ∈ N and α ∈ [0, 1], we have
M Ln,α (q) ⊂ M Ln,0 (q) = SL∗n (q)
.
Remark 3.5 If we consider λ = 1 and n = 0 we obtain the Theorem 3.1
from [2]. Also, for λ = 1 and n ∈ N, we obtain the Theorem 3.3 from [2].
Remark 3.6 If we consider λ = 1 and D = Dβ,γ (see [1] or [2]) in the
above theorem we obtain the Theorem 3.1 from [1].
Theorem 3.2 Let n ∈ N, α ∈ [0, 1] and λ ≥ 1. If F (z) ∈ M Ln,α (q)
then f (z) = La F (z) ∈ SL∗n (q), where La is the Libera-Pascu integral operator
defined by (1).
Proof.
From (1) we have
(1 + a)F (z) = af (z) + zf ′ (z)
and, by using the linear operator Dλn+1 and if we consider f (z) =
we obtain


(1 + a)Dλn+1 F (z) = aDλn+1 f (z) + Dλn+1 z +
=

aDλn+1 f (z)

+z+

∞
X

∞
X

j=2

P∞

j=2

aj z j ,



jaj z j  =

(1 + (j − 1)λ)n+1 jaj z j

j=2

We have (see the proof of the above theorem)

z+

∞
X

j=2

j (1 + (j − 1)λ)n+1 aj z j =


1
λ − 1)Dλn+1 f (z) + Dλn+2 f (z)
λ

14

(4)

M. Acu - On a class of α-convex functions
Thus
(1 + a)Dλn+1 F (z) = aDλn+1 f (z) +


1
λ − 1)Dλn+1 f (z) + Dλn+2 f (z) =
λ

!

λ−1
1
= a+
Dλn+1 f (z) + Dλn+2 f (z)
λ
λ
or



λ(1 + a)Dλn+1 F (z) = (a + 1)λ − 1) Dλn+1 f (z) + Dλn+2 f (z).
Similarly, we obtain



λ(1 + a)Dλn F (z) = (a + 1)λ − 1) Dλn f (z) + Dλn+1 f (z).
Then
Dλn+1 F (z)
=
Dλn F (z)

n+2
Dλ
f (z)
n+1
Dλ
f (z)

·

n+1
Dλ
f (z)
n f (z)
Dλ

+ ((a + 1) λ − 1) ·

((a + 1) λ − 1) +

n+1
Dλ
f (z)
n f (z)
Dλ

n+1
Dλ
f (z)
n f (z)
Dλ

With notation
Dλn+1 f (z)
= p(z), p(0) = 1
Dλn f (z)
we obtain
Dλn+1 F (z)
=
Dλn F (z)

n+2
Dλ
f (z)
n+1
Dλ
f (z)

· p(z) + (a + 1)λ − 1 · p(z)
p(z) + (a + 1)λ − 1

(5)

Also, we obtain
Dλn+2 f (z) Dλn f (z)
1
Dλn+2 f (z)
Dλn+2 f (z)
=
=
·
·
Dλn f (z) Dλn+1 f (z)
p(z) Dλn f (z)
Dλn+1 f (z)
We have
Dλn+2 f (z)
Dλn f (z)

z+
=

∞
P

(1 + (j − 1)λ)n+2 aj z j

j=2
∞
P

z+

j=2

and

15

(1 + (j − 1)λ)n aj z j

(6)

M. Acu - On a class of α-convex functions

′

zp (z) =



′

z Dλn+1 f (z)
Dλn f (z)

z 1+

∞
P

Dλn+1 f (z) z (Dλn f (z))′
−
·
=
Dλn f (z)
Dλn f (z)
n+1

(1 + (j − 1)λ)

j=2

=

−p(z) ·



z 1+

jaj z j−1

!

Dλn f (z)

−

n
j−1
j=2 (1 + (j − 1)λ) jaj z

P∞

Dλn f (z)



or

zp′ (z) =

z+

n+1

P∞

j=2 j (1 + (j − 1)λ)
Dλn f (z)

aj z

z+

j

∞
P

j=2

−p(z)·

j (1 + (j − 1)λ)n aj z j
Dλn f (z)

.
(7)

By using (4) and (7) we obtain
1
zp (z) =
λ
′

(λ − 1)Dλn+1 f (z) + Dλn+2 f (z)
(λ − 1)Dλn f (z) + Dλn+1 f (z)
−
p(z)
Dλn f (z)
Dλn f (z)

1
Dλn+2 f (z)
=
− p(z)((λ − 1) + p(z)) =
(λ − 1)p(z) +
λ
Dλn f (z)
!

1
=
λ

Dλn+2 f (z)
− p(z)2
n
Dλ f (z)

Thus
λzp′ (z) =
or

!

Dλn+2 f (z)
− p(z)2
Dλn f (z)

Dλn+2 f (z)
= p(z)2 + λzp′ (z).
Dλn f (z)
From (6) we obtain
1
Dλn+2 f (z)
(p(z)2 + λzp′ (z)).
=
n+1
p(z)
Dλ f (z)
16

!

=

M. Acu - On a class of α-convex functions
Then, from (5), we obtain
p(z)2 + λzp′ (z) + ((a + 1)λ − 1) p(z)
zp′ (z)
Dλn+1 F (z)
=
=
p(z)+λ
Dλn F (z)
p(z) + ((a + 1)λ − 1)
p(z) + ((a + 1)λ − 1)
where α ∈ C, Rea ≥ 0 and λ ≥ 1.
Dn+1 F (z)
If we denote Dλ n F (z) = h(z), with h(0) = 1, we have from F (z) ∈ M Ln,α (q)
λ
(see the proof of the above Theorem):
Jn,λ (α, F ; z) = h(z) + α · λ ·

zh′ (z)
≺ q(z)
h(z)

Using the hypothesis, from Theorem 2.1, we obtain
h(z) ≺ q(z)
or
p(z) + λ

zp′ (z)
≺ q(z).
p(z) + ((a + 1)λ − 1)

By using the Theorem 2.1 and the hypothesis we have
p(z) ≺ q(z)
or

Dλn+1 f (z)
≺ q(z).
Dλn f (z)
This means f (z) = Lα F (z) ∈ SL∗n (q) .

Remark 3.7 If we consider λ = 1 and n = 0 we obtain the Theorem 3.2
from [2]. Also, for λ = 1 and n ∈ N, we obtain the Theorem 3.4 from [2].
Remark 3.8 If we consider λ = 1 and D = Dβ,γ ( see [1] or [2]) in the
above theorem we obtain the Theorem 3.2 from [1].
References

17

M. Acu - On a class of α-convex functions
[1]M. Acu, On a subclass of α-uniform convex functions, Archivum Mathematicum, no. 2, Tomus 41/2005, (to appear).
[2]M. Acu, Some subclasses of α-uniformly convex functions, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, Vol. 21(2005), (to appear).
[3]M. Acu, On a subclass of n-starlike functions, (to appear).
[4]M. Acu, On a subclass of n-convex functions, (to appear).
[5]F.M. Al-Oboudi, On univalent funtions defined by a generalized S’al’agean
operator, Ind. J. Math. Math. Sci. 2004, no. 25-28, 1429-1436.
[6]P. Duren, Univalent functions, Springer Verlag, Berlin Heildelberg, 1984.
[7]S. S. Miller and P. T. Mocanu, Differential subordination and univalent
functions, Mich. Math. 28(1981), 157-171.
[8]S. S. Miller and P. T. Mocanu, Univalent solution of Briot-Bouquet differential equation, J. Differential Equations 56(1985), 297-308.
[9]S. S. Miller and P. T. Mocanu, On some classes of first-order differential
subordination, Mich. Math. 32(1985), 185-195.
[10]Gr. S˘al˘agean, On some classes of univalent functions, Seminar of geometric function theory, Cluj-Napoca, 1983.
Mugur Acu
University ”Lucian Blaga” of Sibiu
Department of Mathematics
Str. Dr. I. Rat.iu, No. 5-7
550012 - Sibiu, Romania
E-mail address: acu mugur@yahoo.com

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