Table of Contents
ISRN Mathematical Analysis
Volume 2012, Article ID 106079, 13 pages
http://dx.doi.org/10.5402/2012/106079
Research Article

Inclusion Relationships for Certain Subclasses of Meromorphic Functions Defined by Using the Extended Multiplier Transformations

Department of Mathematics, Faculty of Science (Damietta Branch), Mansoura University, New Damietta 34517, Egypt

Received 22 May 2011; Accepted 4 July 2011

Academic Editors: A. Belhaj and A. Masiello

Copyright © 2012 R. M. El-Ashwah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let βˆ‘ denote the class of analytic functions in the punctured unit disc π‘ˆ βˆ— = { 𝑧 ∢ 0 < | 𝑧 | < 1 } . Set πœ‘ π‘š πœ† , β„“ βˆ‘ ( 𝑧 ) = 1 / 𝑧 + ∞ π‘˜ = 0 [ β„“ + πœ† ( π‘˜ + 1 ) / β„“ ] π‘š 𝑧 π‘˜ ( π‘š ∈ β„• 0 ; β„“ > 0 ; πœ† β‰₯ 0 ; 𝑧 ∈ π‘ˆ βˆ— ) , and define πœ‘ π‘š , πœ‡ πœ† , β„“ in terms of the Hadamard product by πœ‘ π‘š πœ† , β„“ ( 𝑧 ) βˆ— πœ‘ π‘š , πœ‡ πœ† , β„“ ( 𝑧 ) = 1 / 𝑧 ( 1 βˆ’ 𝑧 ) πœ‡ ( πœ‡ > 0 ; 𝑧 ∈ π‘ˆ βˆ— ) . In this paper, we introduce several new subclasses of analytic functions defined by means of the operator 𝐼 π‘š πœ‡ ( πœ† , β„“ ) 𝑓 ( 𝑧 ) = πœ‘ π‘š , πœ‡ πœ† , β„“ βˆ‘ ( 𝑧 ) βˆ— 𝑓 ( 𝑧 ) ( 𝑓 ∈ ; π‘š ∈ β„• 0 ; β„“ > 0 ; πœ† β‰₯ 0 ; πœ‡ > 0 ) . Inclusion properties of these classes and some applications involving integral operator are also considered.