Abstract

In our present investigation, motivated from Noor work, we define the class () of functions of bounded radius rotation of complex order with respect to symmetrical points and learn some of its basic properties. We also apply this concept to define the class . We study some interesting results, including arc length, coefficient difference, and univalence sufficient condition for this class.

1. Introduction

Let denote the class of analytic function satisfying the condition , in the open unit disc and in more simple form: By , , and , we means the well-known subclasses of which consists of univalent, convex, and starlike functions, respectively. In [1], Sakaguchi introduced the class of starlike functions with respect to symmetrical points and is defined as follows: a function given by (1.1) belongs to the class , if and only if Motivated from Sakaguchi work, Das and Singh [2] extend the concepts of to other class in , namely, convex functions with respect to symmetrical points. Let denote the class of convex functions with respect to symmetrical points and satisfying the following condition: Let ,  , be the class of functions analytic in with and This class was introduced in [3]. For , we obtain the class defined by Pinchuk [4], and for , the class reduces to the class of functions with positive real part.

Now, with the help of the aforementioned concepts, we define the class of functions of bounded radius rotation of complex order with respect to symmetrical points as follows.

Definition 1.1. Let in . Then , if and only if where and .
Using the class , we define the class as follows.

Definition 1.2. Let in . Then , if and only if there exists such that where , , and .
It is noticed that, by giving specific values to ,  ,  , and in and , we obtain many well-known as well as new subclasses of analytic and univalent functions; for details see [5–11].
Throughout this paper, we will assume, unless otherwise stated, that , , , and .

Lemma 1.3. Let be analytic in where belongs to Then (see [8, 12]).

Lemma 1.4. Let be univalent function in . Then there exists with such that for all , , (see [13]).

2. Some Properties of the Classes

Theorem 2.1. Let . Then the odd function belongs to in .

Proof. Let and consider From logarithmic differentiation of the previous relation, we have or, equivalently, with belongs to . Since is a convex set, we have and hence .

Theorem 2.2. Let . Then

Proof. Let . Then by definition we have Simple computation yields us Using (2.8) in (2.9), we can easily obtain (2.7).
If we put and in Theorem 2.1, we obtain the integral representation for given by Stankiewiez in [14].

Theorem 2.3. Let . Then The function defined by shows that this bound is sharp.

Proof . Since , there exists an odd function with such that with . Let Then (2.13) implies that Equating the coefficients of , we have , and so where we have used the coefficient bounds for the class .

Corollary 2.4. The range of every univalent function contains the disc

Proof . The Koebe one-quarter theorem states that each omitted value of the univalent function of the form (1.1) satisfies Using (2.18) and Theorem 2.3, we obtain the required result.

By using the same method as in [1], we obtain the following result.

Theorem 2.5. Let . Then,  for and ,

Theorem 2.6. Let . Then, for , where and

Proof. We can define, for , , real, the following: The functions are periodic and continuous with period . Since , therefore from (2.22), it follows that we can choose the branches of argument of and as Now we have from (2.22) where is an odd function of the following form: Since , therefore by using Theorem 2.5, we have From (2.22), (2.23), (2.24), and (2.27), we have Moreover, from (2.22) Therefore

Theorem 2.7. Let Then for , where and is a constant depending upon , and only.

Proof. We know that Since , therefore By Theorem 2.1, we have, for , the odd function . This implies that Therefore, we have Since , therefore we have for odd functions ,  , Now using Cauchy Schwarz inequality, we have By Lemma 1.3 and distortion results for the class with a subordination result, we obtain Similarly for we have

Theorem 2.8. Let . Then for where andare the same as in Theorem 2.7 and is a constant depending upon , and only.

Proof. Since, with Cauchy theorem gives therefore Now using Theorem 2.7 for we have Putting we have Similarly we obtain the required result for .

Theorem 2.9. Let .  Then for ,

Proof. We know that for and , Since , therefore By Theorem 2.1, we have, for , the odd function . This implies that Thus, for and , we have Since , therefore we have for odd functions ,  , By using Lemma 1.4, we have Now using Cauchy Schwarz inequality, we have By Lemma 1.3 and distortion result for the class with a subordination result, we obtain Now putting , we obtain Similarly for we have