On Harmonic Functions Defined by Differential Operator with Respect to -Symmetric Points

We introduce new classes and of harmonic univalent functions with respect to -symmetric points defined by differential operator. We determine a sufficient coefficient condition, representation theorem, and distortion theorem. 1. Introduction A continuous function is a complex valued harmonic function in a complex domain if both and are real harmonic in . In any simply connected domain we can write , 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 in . See Clunie and Shell-Small (see [1]). Thus for , we may write Note that reduces to , the class of normalized analytic univalent functions if the coanalytic part of is identically zero. Also, denote by the subclasses of consisting of functions that map onto starlike domain. A function is said to be starlike of order in denoted by (see [2]) if A function of normalized univalent analytic functions is said to be starlike with respect to symmetrical points in if it satisfies this class was introduced and studied by Sakaguchi in 1959 [3]. Some related classes are studied by Shanmugam et al. [4]. In 1979, Chand and Singh [5] defined the class of starlike functions with respect to -symmetric points of order ？？( ). Related classes are also studied by das and Singh [6]. Ahuja and Jahangiri [7] discussed the class which denotes the class of complex-valued, sense-preserving, harmonic univalent functions of the form (1) and satisfying the condition In [8], the authors introduced and studied the class which denotes the class of complex-valued, sense-preserving, harmonic univalent functions of the form (1) and where From the definition of we know The differential operator was introduced by Ali Abubaker and Darus [9]. We define the differential operator of the harmonic function given by (5) as where and also , , , for , and is the Pochhammer symbol defined by We note that when , ,？and we obtain the Ruscheweyh derivative for harmonic functions (see [7]), when we obtain the Salagean operator for harmonic functions (see [10]), and when , we obtain the operator for harmonic functions given by Al-Shaqsi and Darus [11]. Let denote the class of complex-valued, sense-preserving, harmonic univalent functions of the form (5) which satisfy the condition where , , , and the functions and are of the form Further, denote by the subclasses of , such that the functions and in are of the form and the functions and in are of the form In this paper, we obtain inclusion properties and coefficient conditions for