Paper-02-22-2010. 101KB Jun 04 2011 12:03:08 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 22/2010
pp. 17-22

ON A SUBCLASS OF ANALYTIC FUNCTIONS DEFINED BY
RUSCHEWEYH DERIVATIVE AND GENERALIZED SALAGEAN
OPERATOR

Alina Alb Lupaş
P
j
Abstract. Let A (p, n) = {f ∈ H(U ) : f (z) = z p + ∞
j=p+n aj z , z ∈ U },
n : A → A, defined
with A (1, 1) = A. We consider in this paper the operator RDλ,γ


n
n
n

n
by RDλ,γ f (z) := (1 − γ) R f (z) + γDλ f (z), where Dλ f (z) = Dλ Dλn−1 f (z) is
the generalized Sălăgean operator and (n + 1)Rn+1 f (z) = z(Rn f (z))′ + nRn f (z),
n ∈ N0 , N0 = N ∪ {0} is the Ruscheweyh operator. By making use of the above
mentioned differential operator, a new subclass of univalent functions in the open
unit disc is introduced. The new subclass is denoted by RDλ (n, µ, α, λ). 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
Denote by U the unit disc of the complex plane, U = {z ∈ C : |z| < 1} and
H(U ) the space of holomorphic functions in U .
Let
∞
P
aj z j , z ∈ U },
A (p, n) = {f ∈ H(U ) : f (z) = z p +
j=p+n

with A (1, n) = An , A (1, 1) = A1 = A and

H[a, n] = {f ∈ H(U ) : f (z) = a + an z n + an+1 z n+1 + . . . , z ∈ U },
where p, n ∈ N, a ∈ C.
Let S denote the subclass of functions that are univalent in U .
By S ∗ (α) we denote a subclass of A consisting of starlike univalent functions of
order α, 0 ≤ α < 1 which satisfies
 ′ 
zf (z)
> α, z ∈ U.
(1)
Re
f (z)
17

Alb Lupas Alina - On a subclass of analytic functions defined by...

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,
(2)
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 A which satisfy
Re f ′ (z) > α,

z ∈ U.

(3)

It is well known that K(α) ⊂ S ∗ (α) ⊂ S.
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 ).
Definition 1. (Al Oboudi [5]) For f ∈ A, λ ≥ 0 and n ∈ N, the operator Dλn is
defined by Dλn : A → A,

Dλ0 f (z) = f (z)
Dλ1 f (z) = (1 − λ) f (z) + λzf ′ (z) = Dλ f (z)
...
Dλn+1 f (z)

= (1 − λ) Dλn f (z) + λz (Dλn f (z))′ = Dλ (Dλn f (z)) , for z ∈ U.
P
j
Remark 1. If f ∈ A and f (z) = z + ∞
j=2 aj z , then
Dλn f (z) = z +

for z ∈ U .

∞
P

[1 + (j − 1) λ]n aj z j ,

j=2

Remark 2. For λ = 1 in the above definition we obtain the Sălăgean differential
operator [10].
Definition 2. (Ruscheweyh [9]) For f ∈ A and n ∈ N, the operator Rn is
defined by Rn : A → A,
R0 f (z) = f (z)
R1 f (z) = zf ′ (z)
...
(n + 1) R

n+1

f (z) = z (Rn f (z))′ + nRn f (z) , for z ∈ U.
18

Alb Lupas Alina - On a subclass of analytic functions defined by...

P
P∞
j

n
n
j
Remark 3. If f ∈ A and f (z) = z+ ∞
j=2 aj z , then R f (z) = z+ j=2 Cn+j−1 aj z ,
for z ∈ U .
To prove our main theorem we shall need the following lemma.
Lemma 1. [8] Let p be analytic in U with p(0) = 1 and suppose that


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

(4)

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

Definition 3. For a function f ∈ A we define the differential operator
n
RDλ,γ
f (z) = (1 − γ) Rn f (z) + γDλn f (z),

(5)

where n ∈ N0 , N0 = N ∪ {0}.
Remark 4. For λ = 1 the above defined operator was introduced in [1].
Definition 4. We say that a function f ∈ A is in the class RDγ (n, µ, α, λ),
n ∈ N, µ ≥ 0, α ∈ [0, 1), γ ≥ 0 if


!µ



n+1
f (z)
 RDλ,γ

z

 < 1 − α,
z ∈ U.
(6)
−
1
n f (z)


z
RDλ,γ


Remark 5. The family RDγ (n, µ, α, λ) is a new comprehensive class of analytic functions which includes various new classes of analytic univalent functions
as well as some very well-known ones. For example, RD1 (n, µ, α, λ) was studied in [6], RD0 (n, µ, α, λ) was studied in [3], RDγ (n, µ, α, 1) was studied in [4],

RD1 (0, 1, α, 1) = S ∗ (α) , RD1 (1, 1, α, 1) = K (α) and RD1 (0, 0, α, 1) = R (α). Another interesting subclass is the special case RD1 (0, 2, α, 1)=B (α) which has been
introduced by Frasin and Darus [7] and also the class RD1 (0, µ, α, 1) = B(µ, α)
which has been introduced by Frasin and Jahangiri [8].

19

Alb Lupas Alina - On a subclass of analytic functions defined by...

In this note we provide a sufficient condition for functions to be in the class
RDγ (n, µ, α, λ). Consequently, as a special case, we show that convex functions of
order 1/2 are also members of the above defined family.
Theorem 2. For the function f ∈ A, n ∈ N, µ ≥ 0, 1/2 ≤ α < 1 if
(n + 2)


1
γ n+2−
λ



n+2
RDλ,γ
f (z)
n+1
RDλ,γ
f (z)

− µ (n + 1)

n+1
RDλ,γ
f (z)
n f (z)
RDλ,γ

−




Dλn+2 f (z) − Dλn+1 f (z)
1 Dλn+1 f (z) − Dλn f (z)
+ µγ n + 1 −
n f (z)
n+1
λ
RDλ,γ
RDλ,γ
f (z)
+ (µ − 1) (n + 1) ≺ 1 + βz, z ∈ U,

where
β=

(7)

3α − 1
,
2α

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

n+1
f (z)
RDλ,γ

z

z

n f (z)
RDλ,γ

!µ

,

(8)

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



1
γ n+2−
λ



(9)



Dλn+2 f (z) − Dλn+1 f (z)
1 Dλn+1 f (z) − Dλn f (z)
+ µγ n + 1 −
n f (z)
n+1
λ
RDλ,γ
RDλ,γ
f (z)
+µ (n + 1) − (n + 2) .

Using (7) we get


zp′ (z)
Re 1 +
p(z)
Thus, from Lemma 1 we deduce that

 RDn+1 f (z)
λ,γ
Re

z



>

3α − 1
.
2α

z
n f (z)
RDλ,γ
20

!µ 



> α.

Alb Lupas Alina - On a subclass of analytic functions defined by...

Therefore, f ∈ RDγ (n, µ, α, λ), by Definition 4.
As a consequence of the above theorem we have the following interesting corollaries [2].
Corollary 3. If f ∈ A and
(
)
′′
′
2zf (z) + z 2 f ′′ (z) zf ′′ (z)
1
− ′
Re
>− ,
′
′′
f (z) + zf (z)
f (z)
2
then



zf ′′ (z)
1
Re 1 + ′
> ,
f (z)
2

z ∈ U,

z ∈ U.

(10)

(11)



That is, f is convex of order 12 , or f ∈ RD1 1, 1, 21 , 1 .
Corollary 4. If f ∈ A and
(
)
′
′
2zf ′ (z) + z 2 f ′′ (z)
1
Re
>− ,
′
′′
f (z) + zf (z)
2



then f ∈ RD1 1, 0, 12 , 1 , that is


1
Re f ′ (z) + zf ′′ (z) > ,
2
and


zf ′′ (z)
1
Re 1 + ′
> ,
f (z)
2

z ∈ U,

z ∈ U.

(12)

(13)

Corollary 5. If f ∈ A

then

1
Re f ′ (z) > ,
2

z ∈ U,

z ∈ U.

(15)

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


≡ R 21 .
Corollary 6. If f ∈ A and

 ′′
3
zf (z) zf ′ (z)
−
>− ,
Re
′
f (z)
f (z)
2
21

(14)

1
2

then f ∈ RD1 (0, 0, 12 , 1)

z ∈ U,

(16)

Alb Lupas Alina - On a subclass of analytic functions defined by...

then f is starlike of order 12 , hence f ∈ RD1 (0, 1, 21 , 1).

References
[1] A. Alb Lupaş, On special differential subordinations using Sălăgean and Ruscheweyh operators, Mathematical Inequalities and Applications, Volume 12, Issue
4, October 2009 (to appear).
[2] A. Alb Lupaş, A subclass of analytic functions defined by Ruscheweyh derivative, Acta Universitatis Apulensis, nr. 19/2009, 31-34.
[3] A. Alb Lupaş and A. Cătaş, On a subclass of analytic functions defined
by Ruscheweyh derivative, Analele Universităţii din Oradea, Fascicula Matematica, Tom XVI (2009), 225-228.
[4] A. Alb Lupaş and A. Cătaş, On a subclass of analytic functions defined by a
generalized Sălăgean and Ruscheweyh operators, J ournal of Approximation Theory
and Applications, Vol. 4, No. 1-2 (2008), 1-5.
[5] F.M. Al-Oboudi, On univalent functions defined by a generalized Sălăgean
operator, Ind. J. Math. Math. Sci., 27 (2004), 1429-1436.
[6] A. Cătaş and A. Alb Lupaş, On a subclass of analytic functions defined by a
generalized Sălăgean operator, Romai Journal, vol. 4, nr. 2(2008), 57-60.
[7] B.A. Frasin and M. Darus, On certain analytic univalent functions, Internat.
J. Math. and Math. Sci., 25(5), 2001, 305-310.
[8] B.A. Frasin and Jay M. Jahangiri, A new and comprehensive class of analytic
functions, Analele Universităţii din Oradea, Tom XV, 2008, 61-64.
[9] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math.
Soc., 49(1975), 109-115.
[10] G. St. Sălăgean, Subclasses of univalent functions, Lecture Notes in Math.,
Springer Verlag, Berlin, 1013(1983), 362-372.
Alb Lupaş Alina
Department of Mathematics
University of Oradea
str. Universităţii nr. 1, 410087, Oradea, Romania
email: dalb@uoradea.ro

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