Multiplier family of harmonic univalent functions R.A. Al-Khal Department of Mathematics, Faculty of Science, Girls College, P.O. Box 838, Dammam, Saudi Arabia article info Keywords: Harmonic functions Generalized Bernardi–Libera–Livingston integral operator Distortion theorems Closure properties Convolution Neighborhoods abstract The aim of this paper is to study a multiplier family of harmonic univalent functions using the sequences fc n g and fd n g of positive real numbers. By specializing fc n g and fd n g, the generalized Bernardi–Libera–Livingston integral operator is modified for such functions and the closure of the multiplier family under the modified integral operator is determined. Also, convolution products, closure properties, distortion theorems, convex combinations and neighborhoods for such functions are given. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction Let S H denote the class of functions f which are complex-valued, harmonic, univalent, sense-preserving in the open unit disk U normalized by f ð0Þ¼ f z ð0Þ 1 ¼ 0. Each f 2 S H can be expressed as f ¼ h þ  g, where hðzÞ¼ z þ X 1 n¼2 a n z n ; gðzÞ¼ X 1 n¼1 b n z n ; jb 1 j < 1 ð1Þ are analytic in U. A necessary and sufficient condition for f to be locally univalent and sense-preserving in U is that jh 0 ðzÞj > jg 0 ðzÞj in U. Clunie and Sheill-Small [2] studied the class S H with some geometric subclasses of S H . Let HPðbÞ denote the subclass of S H satisfying Refh 0 ðzÞþ g 0 ðzÞg > b; 0 6 b < 1, which was studied by Yalçin et al. [7]; they also denote by HP  ðbÞ the subclass of HPðbÞ such that the functions h and g in f ¼ h þ  g are of the form hðzÞ¼ z  X 1 n¼2 ja n jz n ; gðzÞ¼ X 1 n¼1 jb n jz n ; jb 1 j < 1: ð2Þ It is known [7] that X 1 n¼2 nja n jþ X 1 n¼1 njb n j 6 1  b; ða 1 ¼ 1 and 0 6 b < 1Þ: ð3Þ Then f 2 HPðbÞ. Condition (3) is also necessary if f 2 HP  ðbÞ. A function f ¼ h þ  g, where h and g are given by (2), is said to be in the multiplier family FH b ðfc n g; fd n gÞ if there exist sequences fc n g and fd n g of positive real numbers such that X 1 n¼2 c n ja n jþ X 1 n¼1 d n jb n j 6 1  b; d 1 jb 1 j < 1: ð4Þ 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.08.022 E-mail address: ranaab@hotmail.com Applied Mathematics and Computation 215 (2009) 2238–2242 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc