Mugur Acu. 141KB Jun 04 2011 12:09:15 AM

Mugur Acu - On a subclass of α-uniform convex functions

ON A SUBCLASS OF α-UNIFORM
CONVEX FUNCTIONS
by
Mugur Acu
Abstract. In this paper we define a subclass of α -uniform convex functions by using the
Sălăgean differential operator and we obtain some properties of this class.
2000 Mathematics Subject Classifications: 30C45
Key words and Phrases: Libera type integral operator, α -uniform convex functions, Sălăgean
differential 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 consider the integral operator La : A → A defined as:

1+ a
f(z) = LaF(z) = a

z

∫ F(t )t
t

a −1

dt , a ∈ C , Re(a) ≥ 0.

(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. In the form (1) was used first
time by N. N. Pascu.
Let Dn be the Sălăgean differential operator (see [8]) defined as:
Dn : A → A , n ∈ N and D0f(z) = f(z)
1
D f(z) = Df(z) = z f’(z) , Dnf(z) = D(Dn-1f(z))

2. Preliminary results

Definition 2.1 [3] Let f ∈ A . We say that f is n-uniform starlike function of order
and type if

where

≥ 0,

 D n +1f (z) 
D n +1f (z)
 ≥
−1 + , z ∈ U
n
n

D
f
(

z
)
D
f
(
z
)


∈ [-1, 1), + ≥ 0 , n ∈ N. We denote this class with USn( , ).

Re 

3

Mugur Acu - On a subclass of α-uniform convex functions

Remark 2.1 Geometric interpretation: f ∈ USn( , ) if and only if

D n+1f (z)

take
D n f (z)

all values in the convex domain included in right half plane D , where D , is a
elliptic region for > 1, a parabolic region for = 1, a hyperbolic region for 0 <
< 1, the half plane u > for = 0.

Remark 2.2 If we take n = 0 and = 1 in definition 2.1 we obtain US0(1, ) = SP((1
– )/2 , (1+ )/2), where the class SP( α, ) was introduced by F. Ronning in [9]. Also
we have USn( , ) ⊂ S*, where S* is the well know class of starlike functions.

Definition 2.2 [3] Let f ∈ A. We say that f is α -uniform convex function, α ∈ [0, 1]
if



Re (1 − α)

 zf ' ' (z) 
zf ' (z)


+ α1 +
f ' (z) 
f (z)



where z ∈ U. We denote this class with UMα.

≥

 zf ' (z) 
zf ' ' (z)
,
− 1 + α
(1 − α)
f ' (z)
 f (z)


Remark 2.3 Geometric interpretation: f ∈ UMα if and only if

 zf ' ' (z) 
zf ' (z)

+ α1 +
f ( z)
f ' (z) 

take all values in the parabolic region Ω = {w : |w – 1| ≤ Re(w) } = {w =u + iv; v2 ≤
2u – 1}. Also we have UMα ⊂ Mα, where Mα is the well know class of α -convex
J( α , f; z) = (1 − α)

functions introduced by P. T. Mocanu in [7].

4

Mugur Acu - On a subclass of α-uniform convex functions

The next theorem is result of the so called ”admissible functions method” introduced

by P. T. Mocanu and S. S. Miller (see [4], [5], [6]).

Theorem 2.1 Let h convex in U and Re[ h(z) + ] > 0, z ∈ U. If p ∈ H(U) with p(0)
= h(0) and p satisfied the Briot-Bouquet differential subordination p(z) +

zp' (z)
p h(z), then p(z) p h (z ).
p( z ) + δ
3. Main results

Definition 3.1 Let α ∈ [0, 1] and n ∈ N. We say that f ∈ A is in the class UDn,α( ,
), ≥ 0, ∈ [-1, 1], + ≥ 0 if



Re (1 − α)



D n +1f (z)

D n + 2 f (z) 
α
+
 ≥
D n f (z)
D n +1f (z) 

(1 − α)

D n +1f (z)
D n + 2 f (z)
α
+
−1 +
D n f (z)
D n +1f (z)

Remark 3.1 We have UDn,0( , ) = USn( , ) ⊂ S*, UD0,α(1, 0) = UMα and UD0,1( ,

 + 

 where USc( , ) is the class of the uniform convex
 +1 

) = USc( , ) ⊂ Sc 

functions of type and order introduced by I. Magdaş in [2] and Sc(δ)is the well
know class of convex functions of order δ.

Remark 3.2 Geometric interpretation: f ∈ UDn,α( , ) if and only if
Jn(α, f; z ) = (1 − α)

D n +1f (z)
D n + 2 f (z)
α
+
D n f (z)
D n +1f (z)

take all values in the convex domain included in right half plane D , , where D
defined in remark 2.1.

,

Theorem 3.1 For all α, α’ ∈ [0, 1] with α < α’ we have UDn,α’( , ) ⊂ UDn,α( , ).

Proof.



From f ∈ UDn,α’( , ) we have

Re (1 − α' )



D n +1f (z)
D n + 2 f (z) 
α
'
+

≥
D n f (z)
D n +1f (z) 

With notations

(1 − α' )

D n +1f (z)
D n + 2 f (z)
α
'
+
−1 +
D n f (z)
D n +1f (z)

D n+1f (z)
= p(z), where p(z) = 1 + p1z + … we have
D n f (z)
5

is

Mugur Acu - On a subclass of α-uniform convex functions
(D n +1f (z))' D n f (z) − D n +1f (z)(D n f (z))' D n+ 2 f (z)  D n +1f (z) 

=
zp’(z) = z
−
(D n f (z)) 2
D n f (z)  D n f (z) 

2

zp' (z) D n + 2 f (z) D n +1f (z)
= n +1
− n
p( z )
D f (z)
D f (z)

and thus we obtain
Jn(α, f; z ) = p(z) + α’·
Now we have p(z) + α ·

zp' (z)
p( z )

zp' (z)
take all values in the convex domain included
p( z )

in right half plane D , .
If we consider h ∈ HU(U), with h(0) = 1, which maps the unit disc U into the
convex domain D , , we have Re(h(z)) > 0 and from hypothesis α’ > 0. From here
follows that Re

1
·h(z) > 0. In this conditions from theorem 2.1, whit δ = 0we obtain
α'

p(z) p h (z ), or p(z) take all values in D , .

If we consider the function g : [0, α] → C, g(u) = p(z) + u ·

zp' (z)
, with g(0)
p( z )

= p(z) ∈ D , and g(α’) ∈ D , . Since the geometric image of g(α) is on the segment
obtained by the union of the geometric image of g(0) and g(α’), we have g(α) ∈ D , ,
or

zp' (z)
∈D,.
p( z )
Thus Jn(α, f; z ) take all values in D , , or f ∈ UDn,α( , ).
p(z) + α ·

Remark 3.3 From theorem 3.1 we have UDn,α( , ) ⊂ UDn,0( , ) for all α ∈ [0, 1],
and from remark 3.1 we obtain that the functions from the class UDn,α( , ) are
univalent.
Theorem 3.2 If F(z) ∈ UDn,α( , ) then f(z) = La(F)(z) ∈ USn( , ), where La is the
integral operator defined by (1).
Proof. From (1) we have
(1 + a)F(z) = af(z) + zf’(z)
By means of the application of the linear operator Dn +1 we obtain
(1 +a)Dn +1F(z) = aDn +1F(z) + Dn +1(zf‘(z))
or
(1 +a)Dn +1F(z) = aDn +1F(z) + Dn +2f(z)

6

Mugur Acu - On a subclass of α-uniform convex functions

Thus:

D n + 2 f (z) D n +1f (z)
D n +1f (z)
⋅
+
⋅
a
D n+1F(z) D n + 2 f (z) + aD n +1f (z) D n +1f (z) D n f (z)
D n f (z)
=
=
D n +1f (z)
D n F(z)
D n +1f (z) + aD n f (z)
+a
D n f (z)
With notation

D n+1f (z)
= p(z) where p(z) = 1 + p1z + ... we have:
D n f (z)

 D n +1f (z)  z(D n +1f (z))'⋅D n f (z) − D n +1f (z) ⋅ (D n f (z))'
 =
zp’(z) = z ·  n

D
f
(
z
)
(D n f (z)) 2


,

=
and

D n + 2 f (z) ⋅ D n f (z) − (D n +1f (z)) 2
(D n f (z)) 2

D n + 2 f (z) D n+1f (z) D n + 2 f (z)
1
·zp’(z) = n +1
–
= n +1
– p(z)
p( z )
D f (z)
D n f (z)
D f (z)
It follows:

1
D n + 2 f (z)
· zp(z)
= p(z) +
n +1
p( z )
D f (z)



1
p(z) ⋅  zp' (z) ⋅
+ p(z)  + a ⋅ p(z)
p( z )
1
D F(z)


· zp’(z).
= p(z) +
=
n
p( z ) + a
p( z ) + a
D F(z)
Thus we obtain:

n+1

If we denote

D n+1F(z)
= q(z), with q(0) = 1, and we consider h ∈ HU(U), with
D n F(z)

h(0) = 1, which maps the unit disc U into the convex domain included in right half
plane D , , we have from F(z) ∈ UDn,α( , ) (see remark 3.2):

zq' (z)
p h(z)
q(z)
From theorem 2.1, whit δ = 0 we obtain q(z) p h(z), or
1
p(z) +
· zp(z) p h (z )
p( z ) + a
q(z) + α ·

Using the hypothesis and the construction of the function h(z) we obtain from
theorem 2.1 p(z) p h(z) or f(z) ∈ USn( , ) (see remark 2.1).

7

Mugur Acu - On a subclass of α-uniform convex functions

Remark 3.4 From theorem 3.2 with α = 0 we obtain the theorem 3.1 from [1] which
assert that the integral operator La, defined by (1), preserve the class USn( , ).

References

[1] M. Acu, D. Blezu, A preserving property of a Libera type operator, Filomat,
14(2000),13-18.

[2] A. W. Goodman, On uniformly convex function, Ann. Polon. Math.,
[3]
[4]
[5]
[6]
[7]
[8]

LVIII(1991), 86-92.
I. Magdaş, Doctoral thesis, University ”Babes-Bolyai” Cluj-Napoca, 1999.
S. S. Miller, P. T. Mocanu, Differential subordinations and univalent functions,
Mich. Math., 28(1981), 157-171.
S. S. Miller, P. T. Mocanu, Univalent solution of Briot-Bouquet differential
equations, J. Differential Equations, 56(1985), 297-308.
S. S. Miller, P. T. Mocanu, On some classes of first order differential
subordinations, Mich. Math., 32(1985), 185-195.
P. T. Mocanu, Une propriété de convexité géenéralisée dans la theorie de la
representation conforme, Mathematica (Cluj), 11(34), 1969, 127-133 .
Gr.Sălăgean, On some classes of univalent functions, Seminar of geometric
function theory, Cluj-Napoca ,1983.

[9] F.Ronning, On starlike functions associated with parabolic regions, Ann.
Univ. Mariae Curie-Sklodowska, Sect.A, 45(14), 1991, 117-122.

Author:
Mugur Acu - ”Lucian Blaga” University, Department of Mathematics, 2400 Sibiu,
Romania.

8