General Mathematics Vol. 17, No. 4 (2009), 75–81
A note on a subclass of analytic functions
defined by a generalized Sălăgean and
Ruscheweyh operator
Alina Alb Lupaş, Adriana Cătaş
Abstract
By means of Sălăgean differential operator and Ruscheweyh derivative we define a new class BL(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: Primary 30C45.
Key words and phrases: Analytic function, starlike function, convex
function, Sălăgean operator, Ruscheweyh derivative.
1
Introduction and definitions
Let An denote the class of functions of the form
(1)
∞
X
f (z) = z +
aj z j
j=n+1
75
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Alina Alb Lupaş, Adriana Cătaş
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
µ 0 ¶
zf (z)
(2)
Re
> α, z ∈ U.
f (z)
Further, a function f belonging to S is said to be convex of order α in
U , if and only if
(3)
µ
Re
zf 00 (z)
+1
f 0 (z)
¶
> α,
z ∈ U,
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
Re f 0 (z) > α,
(4)
z ∈ U.
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 ).
Let S m be the Sălăgean differential operator [8], S m : An → An , n ∈ N,
m ∈ N ∪ {0} , defined as
S 0 f (z) = f (z)
S 1 f (z) = Sf (z) = zf 0 (z)
S m f (z) = S(S m−1 f (z)) = z(S m f (z))0 ,
z ∈ U.
A note on a subclass of analytic functions ...
77
In [7] Ruscheweyh has defined the operator Rm : An → An , n ∈ N,
m ∈ N ∪ {0},
R0 f (z) = f (z)
R1 f (z) = zf 0 (z)
(m + 1)Rm+1 f (z) = z [Rm f (z)]0 + mRm f (z),
z ∈ U.
Let Dλm be a generalized Sălăgean and Ruscheweyh operator introduced
by A. Alb Lupaş in [1], Dλm : An → An , n ∈ N, m ∈ N ∪ {0}, defined as
Dλm f (z) = (1 − λ)Rm f (z) + λS m f (z),
z ∈ U, λ ≥ 0.
We note that if f ∈ An , then
∞
X
¡ m
¢
m
m
Dλ f (z) = z +
λj + (1 − λ) Cm+j−1
aj z j ,
z ∈ U, λ ≥ 0.
j=n+1
For λ = 1, we get the Sălăgean operator [8] and for λ = 0 we get the
Ruscheweyh 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
µ
¶
zp0 (z)
3α − 1
(5)
Re 1 +
>
, z ∈ U.
p(z)
2α
Then Re p(z) > α for
2
z ∈ U and 1/2 ≤ α < 1.
Main results
Definition 1 We say that a function f ∈ An is in the class BL(m, µ, α, λ),
n ∈ N, m ∈ N ∪ {0}, µ ≥ 0, λ ≥ 0, α ∈ [0, 1) if
¯
¯
¶µ
¯ Dm+1 f (z) µ
¯
z
¯ λ
¯
(6)
−
1
¯
¯ < 1 − α,
m
¯
¯
z
Dλ f (z)
z ∈ U.
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Alina Alb Lupaş, Adriana Cătaş
Remark 1 The family BL(m, 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, BL(0, 1, α, 1)≡
S ∗ (α) , BL(1, 1, α, 1)≡K (α) and BL(0, 0, α, 1)≡R (α). Another interesting
subclasses are the special case BL(0, 2, α, 1)≡B (α) which has been introduced
by Frasin and Darus [5] , the class BL(0, µ, α, 1) ≡ B(µ, α) introduced by
Frasin and Jahangiri [6], the class BL(m, µ, α, 1) = BS(m, µ, α) introduced
and studied by A.Cătaş and A. Alb Lupaş [4] and the class BL(m, µ, α, 0) =
BR(m, µ, α) introduced and studied by A.Cătaş and A. Alb Lupaş [1].
In this note we provide a sufficient condition for functions to be in the
class BL(m, 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 1 For the function f ∈ An , n ∈ N, m ∈ N ∪ {0}, µ ≥ 0, λ ≥
0, 1/2 ≤ α < 1 if
(7)
(m + 2) (1 − λ) Rm+2 f (z) − (m + 1) (1 − λ) Rm+1 f (z) + λS m+2 f (z)
−
(1 − λ) Rm+1 f (z) + λS m+1 f (z)
(m + 1) (1 − λ) Rm+1 f (z) − m (1 − λ) Rm f (z) + λS m+1 f (z)
µ
+µ ≺ 1+βz, z ∈ U,
(1 − λ) Rm f (z) + λS m f (z)
where
β=
3α − 1
,
2α
then f ∈ BL(m, µ, α, λ).
Proof. If we consider
(8)
Dm+1 f (z)
p(z) = λ
z
µ
z
m
Dλ f (z)
¶µ
,
A note on a subclass of analytic functions ...
79
then p(z) is analytic in U with p(0) = 1. A simple differentiation yields
(9)
zp0 (z)
(m + 2) (1 − λ) Rm+2 f (z) − (m + 1) (1 − λ) Rm+1 f (z) + λS m+2 f (z)
=
−
p(z)
(1 − λ) Rm+1 f (z) + λS m+1 f (z)
µ
(m + 1) (1 − λ) Rm+1 f (z) − m (1 − λ) Rm f (z) + λS m+1 f (z)
− 1 + µ.
(1 − λ) Rm f (z) + λS m f (z)
Using (7) we get
µ
¶
zp0 (z)
3α − 1
Re 1 +
>
.
p(z)
2α
Thus, from Lemma 1 we deduce that
(
µ
¶µ )
z
Dλm+1 f (z)
Re
> α.
z
Dλm f (z)
Therefore, f ∈ BL(m, µ, α, λ), by Definition 1.
As a consequence of the above theorem we have the following interesting
corollaries [2].
Corollary 1 If f ∈ An and
½
¾
9zf 00 (z) + 72 z 2 f 000 (z) 2zf 00 (z)
5
(10)
Re
−
>
−
,
1
0
f (z)
2
f 0 (z) + 2 zf 00 (z)
then
½
(11)
Re
zf 00 (z)
1+ 0
f (z)
¾
3
> ,
7
z ∈ U.
That is, f is convex of order 37 .
Corollary 2 If f ∈ An
(12)
and
½
¾
zf 00 (z)
1
Re 1 + 0
> ,
f (z)
2
z ∈ U,
z ∈ U,
80
Alina Alb Lupaş, Adriana Cătaş
then
(13)
1
Re f 0 (z) > ,
2
z ∈ U.
In another words, if the function f is convex of order
¡ ¢
R 12 .
Corollary 3 If f ∈ An and
½ 00
¾
3
zf (z) zf 0 (z)
>− ,
(14)
Re
−
0
f (z)
f (z)
2
1
2
then f ∈ BL(0, 0, 21 , 1) ≡
z ∈ U,
then
½
(15)
Re
zf 0 (z)
f (z)
¾
1
> ,
2
z ∈ U.
That is, f is a starlike function of order 21 .
Corollary 4 If f ∈ An and
½
¾
2zf 00 (z) + z 2 f 000 (z) zf 00 (z)
1
(16)
Re
− 0
>− ,
0
00
f (z) + zf (z)
f (z)
2
then f ∈ BL(1, 1, 1/2, 1) hence
½
¾
zf 00 (z)
1
(17)
Re 1 + 0
> ,
f (z)
2
z ∈ U,
z ∈ U.
That is, f is convex of order 12 .
References
[1] A. Alb Lupaş, Some diferential subordinations using Sălăgean
and
Ruscheweyh operators, Proceedings of International Conference on Fundamental Sciences, ICFS 2007, Oradea, 58-61.
A note on a subclass of analytic functions ...
81
[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, A note on a subclass of analytic functions
defined by differential Ruscheweyh derivative, Journal of Mathematical
Inequalities, volume 3, Number 3 (2009) (to appear).
[4] A.Alb Lupaş and A. Cătaş, A note on a subclass of analytic functions
defined by differential Sălăgean operator, Buletinul Academiei de Ştiinţe
a Republicii Moldova. Matematica, nr. 2 (60), 2009, 131-134.
[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 Universităţii din Oradea, Tom XV, 2008.
[7] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer.
Math. Soc., 49(1975), 109-115.
[8] G. St. Sălăgean, Subclasses of univalent functions, Lecture Notes in
Math., Springer Verlag, Berlin, 1013(1983), 362-372.
Alina Alb Lupaş, Adriana Cătaş
University of Oradea
Department of Mathematics and Computer Science
1 Universitatii Street, 410087 Oradea, Romania
E-mails: dalb@uoradea.ro , acatas@uoradea.ro