American International Journal of Research in Science, Technology, Engineering & Mathematics
Available online at http://www.iasir.net
ISSN (Print): 2328-3491, ISSN (Online): 2328-3580, ISSN (CD-ROM): 23283629 AIJRSTEM is a refereed, indexed, peer-reviewed, multidisciplinary and open access journal published by International Association of Scientific Innovation and Research (IASIR), USA (An Association Unifying the Sciences, Engineering, and Applied Research)
New Classes Containing Combination of Ruscheweyh Derivative and a New Generalized Multiplier Differential Operator S R Swamy Department of Computer Science and Engineering, R V College of Engineering, Mysore Road, Bangalore-560 059, Karnataka, India Abstract: New classes containing the linear operator obtained as a linear combination of Ruscheweyh derivative and a new generalized multiplier differential operator have been introduced. Sharp results concerning coefficients and distortion theorems of functions belonging to these classes are determined. 2010 Mathematics Subject Classification: 30C45, 30A20, 34A40. Key words and phrases: Analytic function, differential operator, multiplier differential operator.
I.
Introduction
U {z C : z 1}. Let H (U ) be the space of holomorphic functions in U . Let A denote the family of functions in H (U ) of the form Denote by
U
the open unit disc of the complex plane,
f ( z) z ak z k .
(1.1)
k 2
The author has recently introduced the following new generalized multiplier differential operator in [15]. Definition 1.1 Let m N 0 N {0}, 0, a real number such that 0 . Then for f A , a new generalized multiplier operator
I m, was defined by
I 0 , f ( z ) f ( z ) , I 1 , f ( z )
f ( z ) zf ' ( z ) , …, I m, f ( z ) I , ( I m,1 f ( z )) .
Remark 1.2 Observe that for f (z ) given by (1.1), we have
I m, f ( z ) z Ak ( , , m)a k z k ,
(1.2)
k 2
where m
k . Ak ( , , m) m m We note that: i) I1 , , 0 f ( z ) D f ( z ), 0
(1.3) (See
F.
M.
Al-Oboudi
[1]),
ii)
I lm1 , ,0 f ( z ) I lm, f ( z ), l 1, 0 (See A. Catas [3] and he has considered for l 0 ) and iii) I m,1 f ( z ) I m f ( z ), 1 (See Cho and Srivastava [4]) and Cho and Kim [5]). D1m f ( z ) was introduced by Salagean [9] and was considered for Definition
1.3
([8])
m 0 in [2].
For m N 0 , f A ,
the
operator
Rm
is
defined
by
Rm : A A ,
R 0 f ( z ) f ( z ) , R 1 f ( z ) zf ' ( z ) ,…, (m 1) R m 1 f ( z ) z ( R m f ( z )) ' mR m f ( z ), z U . Remark 1.4 If f ( z ) z
a z k A , then R m f ( z ) z k 2 Bk (m)a k z k , z U , where k 2 k
AIJRSTEM 15-781; © 2015, AIJRSTEM All Rights Reserved
Page 74