Applied Mathematics Letters 25 (2012) 952–958 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml On certain subclasses of meromorphic functions associated with certain differential operators E.A. Elrifai, H.E. Darwish ∗ , A.R. Ahmed Department of Mathematics, Faculty of Science, University of Mansoura, Mansoura, Egypt article info Article history: Received 23 May 2011 Accepted 10 November 2011 Keywords: Analytic Meromorphic functions Differential operator Convolution abstract In this work, we study some subordination and convolution properties of certain subclasses of meromorphic functions which are defined by a previously mentioned differential operator. Crown Copyright © 2011 Published by Elsevier Ltd. All rights reserved. 1. Introduction Let Σ denote the class of functions of the form f (z ) = 1 z + ∞  k=1 a k z k (1.1) which are analytic in the punctured unit disk U ∗ := {z : 0 < |z | < 1}= U \{0}, with a simple pole at the origin. If f (z ) and g (z ) are analytic in U , we say that f (z ) is subordinate to g (z ), written f ≺ g or f (z ) ≺ g (z )(z ∈ U ), if there exists a Schwarz function w(z ) in U with w(0) = 0 and |w(z )| < 1 (z ∈ U ), such that f (z ) = g (w(z )) (z ∈ U ). If g (z ) is univalent in U , then the following equivalence relationship holds true: f (z ) ≺ g (z ) (z ∈ U ) ⇐⇒ f (0) = g (0) and f (U ) ⊂ g (U ). For functions f (z ) ∈ Σ given by (1.1) and g (z ) ∈ Σ defined by g (z ) = 1 z + ∞  k=1 a k z k , (1.2) the Hadamard product (or convolution) of f (z ) and g (z ) is given by (f ∗ g )(z ) := 1 z + ∞  k=1 a k b k z k =: (g ∗ f )(z ). (1.3) ∗ Corresponding author. E-mail addresses: Rifai@mans.edu.eg (E.A. Elrifai), Darwish333@yahoo.com (H.E. Darwish), Abdusalam5056@yahoo.com (A.R. Ahmed). 0893-9659/$ – see front matter Crown Copyright © 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2011.11.003