Subclasses of Harmonic Mappings Defined by Convolution

Two new subclasses of harmonic univalent functions defined by using convolution and integral convolution are introduced. These subclasses generate several known and new subclasses of harmonic univalent functions as special cases and provide a unified treatment in the study of these classes. Coefficient bounds, extreme points, distortion bounds, convolution conditions, and convex combination are also determined. 1. Introduction A continuous function is said to be a complex-valued harmonic function in a simply connected domain in complex plane if both real part of and imaginary part of are real harmonic in . Such functions can be expressed as where and are analytic in . We call the analytic part and the coanalytic part of . A necessary and sufficient condition for to be locally univalent and sense-preserving in is that for all in , see [1]. Every harmonic function is uniquely determined by the coefficients of power series expansions in the unit disk given by where for and for . For further information about these mappings, one may refer to [1–5]. In 1984, Clunie and Sheil-Small [1] studied the family of all univalent sense-preserving harmonic functions of the form (1) in , such that and are represented by (2). Note that reduces to the well-known family , the class of all normalized analytic univalent functions given in (2), whenever the coanalytic part of is zero. Let and denote the respective subclasses of and where the images of are convex. Denote by the subclass of for which . The convolution of two functions of the form is given by and the integral convolution is defined by Towards the end of the last century, Jahangiri [3], Silverman [4], and Silverman and Silvia [5] were amongst those who focused on the harmonic starlike functions. Later ？ztürk et. al. [6] defined the class consisting of functions such that and are of the forms which satisfy the condition for some and for all . Several authors [3–16] have investigated various subclasses of harmonic functions. In this work, we introduce a new subclass of harmonic functions defined by convolution. Let be a real constant with , then we denote , the subclass of of functions of the form that satisfy the condition where , , , , and are as given in (3). We also denote , the subclass of of functions of the form that satisfy the condition where is real. We note that the families and are of special interest, because they contain various classes of well-known harmonic univalent functions as well as many new ones. For different choice of , and we obtain the following various classes introduced by other