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 the class . A representation theorem and distortion bounds for the class are also established.

Lemma 1 (see [12]). Let ; if where , then is harmonic, sense-preserving, univalent in and is starlike harmonic of order .

2. Main Results

First, we give a meaningful conclusion about the class .

Theorem 2. Let , where f is given by (1); then defined by (5) is in .

Proof. Let . Then substituting by , where () in (11), respectively, we have According to the definition of and , we know for any and summing up we can get that is, .

Next, a sufficient coefficient condition for harmonic functions in is given.

Theorem 3. Let with and given by (1) and with and given by (5). Let where , , and , for and . Then is sense-preserving harmonic univalent in and .

Proof. Since by Lemma 1, we conclude that is sense-preserving, harmonic univalent, and starlike in . To prove , according to the condition (11), we need to show that where
Using the fact that if and only if , it suffices to show that
On the other hand, for and as given in (20) and (21), respectively, we have Note that, by substituting the value of given by (7) in the previous inequality above, then by (18). Thus concludes the proof of the theorem.

Next, the condition (18) is also necessary for functions in , which is clarified.

Theorem 4. Let with and given by (13) and with and given by (14). Then if and only if where , , and , for and given by (6).

Proof. The if part follows from Theorem 3 upon noting that if the analytic and coanalytic parts of are of the form (13), then . For the only if part, we show that , if the condition (26) does not hold. Thus we can write this is equivalent to That is, The above-required condition must hold for all values of , . Upon choosing the values of on the positive real axis where , we must have If the condition (26) does not hold, then the numerator in (30) is negative for sufficiently close to 1. Hence there exists a in for which the quotient in (30) is negative. This contradicts the required condition for and the proof is complete.

Now, the distortion result is given.

Theorem 5. If , then

Proof. We will only prove the left-hand inequality of the above theorem. The arguments for the right-hand inequality are similar and so we omit it. Let . Taking the absolute value of we obtain by (26): The following covering result follows from the left-hand inequality in Theorem 5.

Corollary 6. If then

Note that other work related to Sakaguchi and classes of functions with respect to symmetric points can be found in [1316].

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

Both authors read and approved the final paper.


The work presented here was partially supported by AP-2013-009 and DIP-2013-001. The authors also would like to thank the referees for the comments given to improve the paper.