IE1art2sec20062. 109KB Jun 04 2011 12:09:02 AM

ACTA UNIVERSITATIS APULENSIS

No 12/2006

SUBCLASSES OF CONVEX FUNCTIONS ASSOCIATED WITH
SOME HYPERBOLA

Mugur Acu
Abstract.In this paper we define some subclasses of convex functions
associated with some hyperbola by using a generalized S˘al˘agean operator and
we give some properties regarding these classes.
2000 Mathematics Subject Classification: 30C45.
Key words and phrases: 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 (see [12]) defined as:
Dn : A → A ,

n ∈ N and D0 f (z) = f (z)

D1 f (z) = Df (z) = zf ′ (z) ,

Remark. If f ∈ S , f (z) = z +
∞
P

j=2

n

j

j aj z .

Dn f (z) = D(Dn−1 f (z)).

∞
P

j=2

aj z j , z ∈ U then Dn f (z) = z +

We recall here the definition of the well - known class of convex functions
!

(

)

zf ′′ (z)
+1 >0 , z ∈U .
CV = S = f ∈ A : Re
f ′ (z)
c

3

M. Acu - Subclasses of convex functions associated with some hyperbola
Let consider the Libera-Pascu integral operator La : A → A defined as:
z

1+aZ
f (z) = La F (z) =
F (t) · ta−1 dt ,
za
0

a∈C,

Re a ≥ 0.

(1)

Generalizations of the Libera-Pascu integral operator was studied by many
mathematicians such are P.T. Mocanu in [8], E. Dr˘aghici in [7] and D. Breaz
in [6].
Definition 1.1.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) .

Remark 1.2. We observe that Dλn is a linear operator and for f (z) =
P
j
z+ ∞
j=2 aj z we have
Dλn f (z)

=z+

∞
X

j=2

(1 + (j − 1)λ)n aj z j .

Also, it is easy to observe that if we consider λ = 1 in the above definition we
obtain the S˘al˘agean differential operator.
The next theorem is result of the so called ”admissible functions method”
introduced by P.T. Mocanu and S.S. Miller (see [9], [10], [11]).
Theorem 1.1. 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),
βp(z) + γ

then p(z) ≺ h(z).

In [4] is introduced the following operator:

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M. Acu - Subclasses of convex functions associated with some hyperbola
Definition 1.2. Let β, λ ∈ R, β ≥ 0, λ ≥ 0 and f (z) = z +
We denote by Dλβ the linear operator defined by

P∞

j=2

aj z j .

Dλβ : A → A ,
Dλβ f (z) = z +

∞
X

j=2

(1 + (j − 1)λ)β aj z j .

Remark 1.3. It is easy to observe that for β = n ∈ N we obtain the
Al-Oboudi operator Dλn and for β = n ∈ N , λ = 1 we obtain the S˘al˘agean
operator Dn .
The purpose of this note is to define some subclasses of convex functions
associated with some hyperbola by using the operator Dλβ defined above and
to obtain some properties regarding these classes.
2. Preliminary results
Definition 2.1. [1] A function f ∈ A is said to be in the class CV H(α)
if it satisfies


)
(
 zf ′′ (z)

√
√

 √
√ zf ′′ (z)



 ′
+
2α
−
2α
2
−
1
+
1
2
2
−
1
+ 2,
 < Re
 f (z)


f ′ (z)

for some α (α > 0) and for all z ∈ U .

Remark 2.1. Geometric interpretation: Let
n

o

Ω(α) = w = u + i · v : v 2 < 4αu + u2 , u > 0 .
Note that Ω(α) is the interior of a hyperbola in the right half-plane which is
symmetric about the real axis and has vertex at the origin. Whit this notations
′′ (z)
+ 1 take all values in the convex
we have f (z) ∈ CV H(α) if and only if zff ′ (z)
domain Ω(α) contained in the right half-plane.
Definition 2.2. [2] Let f ∈ A and α > 0. We say that the function f is
in the class CV Hn (α), n ∈ N , if



(
)
 D n+2 f (z)


√
√
√ Dn+2 f (z)


2 − 1  < Re
2 n+1
2−1 , z ∈U.
 n+1
+
2α
− 2α

D
f (z)

D f (z)

5

M. Acu - Subclasses of convex functions associated with some hyperbola

Remark 2.2. Geometric interpretation: If we denote with pα the analytic and univalent functions with the properties pα (0) = 1, p′ α (0) > 0
and pα (U ) = Ω(α) (see Remark 2.1), then f ∈ CV Hn (α) if and only if
Dn+2 f (z)
≺ pα (z), where the symbol ≺ denotes the subordination in U . We have
Dn+1 f (z)
q

2

− 2α , b = b(α) = 1+4α−4α
and the branch of the square
pα (z) = (1 + 2α) 1+bz
1−z
(1+2α)2
√
√
root w is chosen so that Im w ≥ 0 . If we consider pα (z) = 1 + C1 z + . . . ,
1+4α
.
we have C1 = 1+2α
Theorem 2.1. [2] If F (z) ∈ CV Hn (α), α > 0, n ∈ N , and f (z) =
La F (z), where La is the integral operator defined by (1), then f (z) ∈ CV Hn (α),
α > 0, n ∈ N .
Theorem 2.2. [2] Let n ∈ N and α > 0. If f ∈ CV Hn+1 (α) then
f ∈ CV Hn (α) .
3. Main results
q

Definition 3.1. Let β ≥ 0, λ ≥ 0, α > 0 and pα (z) = (1 + 2α) 1+bz
− 2α ,
1−z
√
1+4α−4α2
where b = b(α) = (1+2α)2 and the branch of the square root w is chosen so
√
that Im w ≥ 0 . We say that a function f (z) ∈ S is in the class CV Hβ,λ (α)
if
Dλβ+2 f (z)
≺ pα (z) , z ∈ U .
Dλβ+1 f (z)
Remark 3.1. Geometric interpretation: f (z) ∈ CV Hβ,λ (α) if and only if

β+2
Dλ
f (z)

β+1
Dλ
f (z)

take all values in the domain Ω(α) which is the interior of a hyperbola

in the right half-plane which is symmetric about the real axis and has vertex at
the origin (see Remark 2.1 and Remark 2.2).
Remark 3.2. We observe that this class generalize the class CV Hn (α)
studied in [2] and the class CV H(α) studied in [1].
Theorem 3.1. Let β ≥ 0, α > 0 and λ > 0 . We have
CV Hβ+1,λ (α) ⊂ CV Hβ,λ (α) .
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M. Acu - Subclasses of convex functions associated with some hyperbola

Proof. Let f (z) ∈ CV Hβ+1,λ (α).
With notation
Dλβ+2 f (z)
, p(0) = 1 ,
p(z) = β+1
Dλ f (z)
we obtain
Dλβ+3 f (z)
Dλβ+3 f (z)
Dλβ+3 f (z) Dλβ+1 f (z)
1
·
= β+1
·
=
p(z) Dλβ+1 f (z)
Dλβ+2 f (z)
Dλ f (z) Dλβ+2 f (z)

(2)

Also, we have
Dλβ+3 f (z)
Dλβ+1 f (z)
and
zp′ (z) =

z+
=
z+



′

z Dλβ+2 f (z)
Dλβ+1 f (z)

z 1+
=

∞
P

j=2

z 1+
−p(z) ·

∞
P

j=2
∞
P

(1 + (j − 1)λ)β+3 aj z j

j=2

(1 + (j − 1)λ)β+1 aj z j

−

Dλβ+2 f (z)
Dλβ+1 f (z)

·

β+2

(1 + (j − 1)λ)



Dλβ+1 f (z)

jaj z

j−1

!

Dλβ+1 f (z)
∞
P

j=2

′

z Dλβ+1 f (z)

β+1

(1 + (j − 1)λ)

jaj z

=

−

j−1

!

Dλβ+1 f (z)

or
z+
zp′ (z) =

∞
P

j=2

j (1 + (j − 1)λ)β+2 aj z j
Dλβ+1 f (z)

z+
− p(z) ·

∞
P

j=2

j (1 + (j − 1)λ)β+1 aj z j
Dλβ+1 f (z)

(3)

We have
z+

∞
X

j=2

j (1 + (j − 1)λ)β+2 aj z j = z +

∞
X

j=2

7

((j − 1) + 1) (1 + (j − 1)λ)β+2 aj z j =

.

M. Acu - Subclasses of convex functions associated with some hyperbola

=z+

∞
X

j=2

= z + Dλβ+2 f (z) − z +
= Dλβ+2 f (z) +
=

∞
X

(1 + (j − 1)λ)β+2 aj z j +

Dλβ+2 f (z)

= Dλβ+2 f (z) −

∞
X

j=2

j=2

(j − 1) (1 + (j − 1)λ)β+2 aj z j =

(j − 1) (1 + (j − 1)λ)β+2 aj z j =

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

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

∞
∞
1X
1X
(1 + (j − 1)λ)β+2 aj z j +
(1 + (j − 1)λ)β+3 aj z j =
λ j=2
λ j=2



1  β+2
1  β+3
Dλ f (z) − z +
Dλ f (z) − z =
λ
λ
z
z
1
1
= Dλβ+2 f (z) − Dλβ+2 f (z) + + Dλβ+3 f (z) − =
λ
λ λ
λ
λ − 1 β+2
1
=
Dλ f (z) + Dλβ+3 f (z) =
λ
λ


1
=
(λ − 1)Dλβ+2 f (z) + Dλβ+3 f (z) .
λ
Similarly we have

= Dλβ+2 f (z) −

z+

∞
X

j=2

j (1 + (j − 1)λ)β+1 aj z j =


1
(λ − 1)Dλβ+1 f (z) + Dλβ+2 f (z) .
λ

From (3) we obtain
1
zp (z) =
λ
′

(λ − 1)Dλβ+2 f (z) + Dλβ+3 f (z)
(λ − 1)Dλβ+1 f (z) + Dλβ+2 f (z)
− p(z)
Dλβ+1 f (z)
Dλβ+1 f (z)

Dλβ+3 f (z)
1
− p(z) ((λ − 1) + p(z)) =
(λ − 1)p(z) + β+1
=
λ
Dλ f (z)
!

1
=
λ

Dλβ+3 f (z)
− p(z)2
Dλβ+1 f (z)
8

!

!

=

M. Acu - Subclasses of convex functions associated with some hyperbola
Thus

Dλβ+3 f (z)
λzp (z) = β+1
− p(z)2
Dλ f (z)
′

or

Dλβ+3 f (z)
= p(z)2 + λzp′ (z) .
β+1
Dλ f (z)
From (2) we obtain

Dλβ+3 f (z)
1 
zp′ (z)
2
′
p(z)
+
λzp
(z)
=
p(z)
+
λ
,
=
p(z)
p(z)
Dλβ+2 f (z)

where λ > 0.
From f (z) ∈ CV Hβ+2,λ (α) we have
p(z) + λ

zp′ (z)
≺ pα (z) ,
p(z)

with p(0) = pα (0) = 1, α > 0, β ≥ 0, λ > 0, and Re pα (z) > 0 from here
construction. In this conditions from Theorem 1.1, we obtain
p(z) ≺ pα (z)
or

Dλβ+2 f (z)
≺ pα (z) .
Dλβ+1 f (z)

This means f (z) ∈ CV Hβ,λ (α) .
Theorem 3.2. Let β ≥ 0, α > 0 and λ ≥ 1 . If F (z) ∈ CV Hβ,λ (α) then
f (z) = La F (z) ∈ CV Hβ,λ (α), 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 Dλβ+2 , we obtain


(1 + a)Dλβ+2 F (z) = aDλβ+2 f (z) + Dλβ+2 z +
9

∞
X

j=2



jaj z j  =

M. Acu - Subclasses of convex functions associated with some hyperbola

= aDλβ+2 f (z) + z +

∞
X

j=2

(1 + (j − 1)λ)β+2 jaj z j

We have (see the proof of the above theorem)
z+

∞
X

j=2

j (1 + (j − 1)λ)β+2 aj z j =


1
(λ − 1)Dλβ+2 f (z) + Dλβ+3 f (z)
λ

Thus
(1 + a)Dλβ+2 F (z) = aDλβ+2 f (z) +


1
(λ − 1)Dλβ+2 f (z) + Dλβ+3 f (z) =
λ

!

1
λ−1
= a+
Dλβ+2 f (z) + Dλβ+3 f (z)
λ
λ
or
λ(1 + a)Dλβ+2 F (z) = ((a + 1)λ − 1) Dλβ+2 f (z) + Dλβ+3 f (z) .
Similarly, we obtain
λ(1 + a)Dλβ+1 F (z) = ((a + 1)λ − 1) Dλβ+1 f (z) + Dλβ+2 f (z) .
Then
Dλβ+2 F (z)
=
Dλβ+1 F (z)

β+3
Dλ
f (z)

β+2
Dλ
f (z)

With notation

·

β+2
Dλ
f (z)

β+1
Dλ
f (z)
β+2
Dλ
f (z)
β+1
Dλ
f (z)

+ ((a + 1)λ − 1) ·

β+2
Dλ
f (z)

β+1
Dλ
f (z)

.

+ ((a + 1)λ − 1)

Dλβ+2 f (z)
= p(z) , p(0) = 1 ,
Dλβ+1 f (z)

we obtain
Dλβ+2 F (z)
Dλβ+1 F (z)

β+3
Dλ
f (z)

=

β+2
Dλ
f (z)

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

We have (see the proof of the above theorem)
Dλβ+3 f (z) Dλβ+2 f (z)
λzp (z) = β+2
·
− p(z)2 =
Dλ f (z) Dλβ+1 f (z)
′

10

(4)

M. Acu - Subclasses of convex functions associated with some hyperbola
Dλβ+3 f (z)
= β+2
· p(z) − p(z)2 .
Dλ f (z)
Thus


1 
Dλβ+3 f (z)
2
′
·
p(z)
+
λzp
(z)
.
=
p(z)
Dλβ+2 f (z)

Then,from (4), we obtain
Dλβ+2 F (z)
p(z)2 + λzp′ (z) + ((a + 1)λ − 1) p(z)
=
=
p(z) + ((a + 1)λ − 1)
Dλβ+1 F (z)
zp′ (z)
= p(z) + λ
,
p(z) + ((a + 1)λ − 1)

where a ∈ C, Re a ≥ 0, β ≥ 0, and λ ≥ 1 .
From F (z) ∈ CV Hβ,λ (α) we have
p(z) +

zp′ (z)
≺ pα (z) ,
1
(p(z) + ((a + 1)λ − 1))
λ

where a ∈ C, Re a ≥ 0, α > 0, β ≥ 0, λ ≥ 1, and from her construction, we
have Re pα (z) > 0. In this conditions we have from Theorem 1.1 we obtain
p(z) ≺ pα (z)
or

Dλβ+2 f (z)
≺ pα (z) .
Dλβ+1 f (z)

This means f (z) = La F (z) ∈ CV Hβ,λ (α) .
Remark 2.3. If we consider β = n ∈ N in the previously results we obtain
the Theorem 3.1 and Theorem 3.2 from [3].

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M. Acu - Subclasses of convex functions associated with some hyperbola
References
[1] M. Acu and S. Owa, Convex functions associated with some hyperbola,
Journal of Approximation Theory and Applications, Vol. 1(2005), (to appear).
[2] M. Acu, On a subclass of n-convex functions associated with some hyperbola, Annals of the Oradea Univ., Fascicola Matematics (2005), (to appear).
[3] M. Acu and S. Owa, On n-convex functions associated with some hyperbola, (to appear).
[4] M. Acu and S. Owa, Note on a class of starlike 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] D. Breaz, Operatori integrali pe spat¸ii de funct¸ii univalente, Editura
Academiei Romˆane, Bucure¸sti 2004.
[7] E. Dr˘aghici, Elemente de teoria funct¸iilor cu aplicat¸ii la operatori integrali univalen’ti, Editura Constant, Sibiu 1996.
[8] P.T. Mocanu, Classes of univalent integral operators, J. Math. Anal.
Appl. 157, 1(1991), 147-165.
[9] S. S. Miller and P. T. Mocanu, Differential subordonations and univalent
functions, Mich. Math. 28 (1981), 157 - 171.
[10] S. S. Miller and P. T. Mocanu, Univalent solution of Briot-Bouquet
differential equations, J. Differential Equations 56 (1985), 297 - 308.
[11] S. S. Miller and P. T. Mocanu, On some classes of first-order differential subordinations, Mich. Math. 32(1985), 185 - 195.
[12] Gr. S˘al˘agean, Subclasses of univalent functions, Complex Analysis. Fifth Roumanian-Finnish Seminar, Lectures Notes in Mathematics, 1013,
Springer-Verlag, 1983, 362-372.
Author:
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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