03-21.2010. 90KB Jun 04 2011 12:07:11 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 22/2010
pp. 35-39

A NOTE ON A SUBCLASS OF ANALYTIC FUNCTIONS DEFINED
˘ AGEAN
˘
BY A GENERALIZED SAL
OPERATOR

˘tas¸
Alina Alb Lupas¸, Adriana Ca
Abstract. By means of a generalized S˘al˘
agean differential operator we define
a new class BO(m, µ, α, λ) involving functions f ∈ An . Parallel results, for some
related classes including the class of starlike and convex functions respectively, are
also obtained.
2000 Mathematics Subject Classification: 30C45
1. Introduction and definitions

Let An denote the class of functions of the form
f (z) = z +

∞
X

aj z j

(1)

j=n+1

which are analytic in the open unit disc U = {z : |z| < 1} and H(U ) the space of
holomorphic functions in U , n ∈ N = {1, 2, ...}.
Let S denote the subclass of functions that are univalent in U .
By S ∗ (α) we denote a subclass of An consisting of starlike univalent functions
of order α, 0 ≤ α < 1 which satisfies
 ′ 
zf (z)
> α, z ∈ U.

(2)
Re
f (z)
Further, a function f belonging to S is said to be convex of order α in U , if and
only if
 ′′

zf (z)
Re
+ 1 > α, z ∈ U
(3)
f ′ (z)

for some α, (0 ≤ α < 1) . We denote by K(α) the class of functions in S which are
convex of order α in U and denote by R(α) the class of functions in An which satisfy
Ref ′ (z) > α,
It is well known that K(α) ⊂ S ∗ (α) ⊂ S.
35

z ∈ U.

(4)

A. Alb Lupa¸s, A. C˘
ata¸s - A note on a subclass of analytic functions defined ...

If f and g are analytic functions in U , we say that f is subordinate to g, written
f ≺ g, if there is a function w analytic in U , with w(0) = 0, |w(z)| < 1, for all z ∈ U
such that f (z) = g(w(z)) for all z ∈ U . If g is univalent, then f ≺ g if and only if
f (0) = g(0) and f (U ) ⊆ g(U ).
Let Dm be a generalized S˘
al˘agean operator introduced by Al-Oboudi in [3],
Dm : An → An , n ∈ N, m ∈ N ∪ {0}, defined as
D0 f (z) = f (z)
D1 f (z) = (1 − λ) f (z) + λzf ′ (z) = Dλ f (z),
m

D f (z) = Dλ (D
We note that if f ∈ An ,

m−1

f (z)),

λ>0

z ∈ U.

then

Dm f (z) = z +

∞
P

[1 + (j − 1) λ]m aj z j , z ∈ U.

j=n+1

For λ = 1, we get the S˘al˘

agean operator [7].
To prove our main theorem we shall need the following lemma.
Lemma 1.[6]Let p be analytic in U with p(0) = 1 and suppose that


zp′ (z)
3α − 1
Re 1 +
>
, z ∈ U.
p(z)
2α
Then Rep(z) > α for

(5)

z ∈ U and 1/2 ≤ α < 1.
2. Main results

Definition 1. We say that a function f ∈ An is in the class BO(m, µ, α, λ),

n ∈ N, m ∈ N ∪ {0}, µ ≥ 0, λ ≥ 0, α ∈ [0, 1) if


µ
 Dm+1 f (z) 

z
 λ

−
1
z ∈ U.
(6)


0, 1/2 ≤ α < 1 if
1 Dλm+2 f (z) µ Dλm+1 f (z) µ − 1
−
+
+ 1 ≺ 1 + βz, z ∈ U

λ Dλm+1 f (z) λ Dλm f (z)
λ
where
β=

(7)

3α − 1
2α

then f ∈ BO(m, µ, α, λ).
Proof. If we consider
p(z) =

Dλm+1 f (z)
z



z

Dλm f (z)

µ

(8)

then p(z) is analytic in U with p(0) = 1. A simple differentiation yields
zp′ (z)
1 Dλm+2 f (z) µ Dλm+1 f (z) µ − 1
−
=
+
.
p(z)
λ Dλm+1 f (z) λ Dλm f (z)
λ

(9)

Using (7) we get



zp′ (z)
Re 1 +
p(z)



>

3α − 1
.
2α

Thus, from Lemma 1 we deduce that
(

µ )
Dλm+1 f (z)
z

> α.
Re
z
Dλm f (z)
Therefore, f ∈ BO(m, µ, α, λ), by Definition 1.
As a consequence of the above theorem we have the following interesting corollaries.
Corollary 3.[1] If f ∈ An and


1
2zf ′′ (z) + z 2 f ′′′ (z) zf ′′ (z)
− ′
>− ,
Re
′
′′
f (z) + zf (z)
f (z)
2
37

z ∈ U,

(10)

A. Alb Lupa¸s, A. C˘
ata¸s - A note on a subclass of analytic functions defined ...

then f ∈ BO(1, 1, 1/2, 1) hence


zf ′′ (z)
1
Re 1 + ′
> ,
f (z)
2

z ∈ U.

(11)

z ∈ U,

(12)

That is, f is convex of order 21 .
Corollary 4.[1] If f ∈ An and


zf ′′ (z)
1
Re 1 + ′
> ,
f (z)
2
then

1
Ref ′ (z) > ,
2

z ∈ U.

(13)

In another
words, if the function f is convex of order


R 21 .

1
2

then f ∈ BO(0, 0, 12 , 1) ≡

Corollary 5.[1] If f ∈ An and


1
f (z) + 3zf ′ (z) + z 2 f ′′ (z) zf ′ (z)
− ′
> ,
Re
′
′
f (z) + zf (z)
f (z)
2

then
Re



zf ′ (z)
f (z)



> 0,

z ∈ U,

z ∈ U.

(14)

(15)

That is, f is a starlike function.
Corollary 6.[1] If f ∈ An and


1
−f (z) + 5zf ′ (z) + z 2 f ′′ (z)
Re
>− ,
f (z) + zf (z)
2
then
Re



f (z)
+ f ′ (z) − 2
z

38



> 2,

z ∈ U.

z ∈ U,

(16)

(17)

A. Alb Lupa¸s, A. C˘
ata¸s - A note on a subclass of analytic functions defined ...

References
[1] A. Alb Lupa¸s, A subclass of analytic functions defined by Ruscheweyh derivative, Acta Universitatis Apulensis, nr. 19/2009, 31-34.
[2] A.Alb Lupa¸s and A. C˘
ata¸s, A note on a subclass of analytic functions defined by differential S˘
al˘
agean operator, Buletinul Academiei de S
¸ tiint¸e a Republicii
Moldova, Matematica, nr. 2(60), 2009, 131-134.
[3] F.M. Al-Oboudi, On univalent functions defined by generalized S˘
al˘
agean operator, Internat. J. Math. and Math. Sci., 27(2004), 1429-1436.
[4] A. C˘
ata¸s and A. Alb Lupa¸s, On Sufficient Conditions for Certain Subclass of
Analytic Functions Defined by Differential S˘
al˘
agean Operator, International Journal
of Open Problem in Complex Analysis, Vol. 1, No. 2, Nov. 2009, 14-18.
[5] B.A. Frasin and M. Darus, On certain analytic univalent functions, Internat.
J. Math. and Math. Sci., 25(5), 2001, 305-310.
[6] B.A. Frasin and Jay M. Jahangiri, A new and comprehensive class of analytic
functions, Analele Universit˘a¸tii din Oradea, Tom XV, 2008, 61-64.
[7] G. St. S˘
al˘agean, Subclasses of univalent functions, Lecture Notes in Math.,
Springer Verlag, Berlin, 1013(1983), 362-372.
Alb Lupa¸s Alina
Department of Mathematics
University of Oradea
str. Universit˘
a¸tii nr. 1, 410087, Oradea, Romania
email: dalb@uoradea.ro
C˘ata¸s Adriana
Department of Mathematics
University of Oradea
str. Universit˘
a¸tii nr. 1, 410087, Oradea, Romania
email: acatas@uoradea.ro

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