We define the classes as follows: if and only if, for , , ; ; , where is a function of bounded boundary rotation. Coefficient estimates, an inclusion result, arclength problem, and some other properties of these classes are studied.

1. Introduction

Let be the class of functions of the form: which are analytic in the open unit disk By , , and denote the subclasses of which are univalent, close-to-convex, starlike, and convex in respectively. Let be the class of functions of bounded boundary rotation. Paate [1] showed that a function defined by (1.1) and , is in if and only if, for , It is geometrically obvious that and

A class of analytic functions related with the class was introduced and studied in [2]. A function is in , if and only if there exists a function such that, for It is clear that

Let denote the class of analytic functions defined by with for

We denote as the class of strongly close-to-convex functions of order in the sense of Pommerenke [3]. A function belongs to if and only if there exists such that for and

Clearly , and when , is a subset of and hence contains only univalent functions. For , can be of infinite valence; see [4].

We now define the following.

Definition 1.1. A function is said to belong to , where is a real number, , , and is called generalized alpha-close-to-convex with argument if and only if there exists such that In (1.4), we choose this branch of argument which equals , , when We note that the condition implies that is nonempty. From the normalization conditions it follows from Definition 1.1 that and therefore Also, it follows from (1.4) that if then for Condition (1.4) is equivalent to the following if and only if there exists such that

We define the class of generalized -close-to-convex functions as If in (1.6), then the class is identical with the class and is the class of close-to-convex functions. Also in the class of close-to-convex function with argument was defined by Goodman and Saff [5]. For details of special cases of with in (1.4), we refer to [6]. The special case with , and in (1.4) leads to the class of functions convex in the direction of the imaginary axis having special normalization; see [7].

2. Main Results

We now prove the main results as follows.

Theorem 2.1. Let Then where The constant cannot be smaller.

Proof. We will use an extended version of the method given in [8] to prove this result.
For the result is obvious. Let By (1.4), (1.5), and (1.6), then there exists a function and a function , such that Let , Then we have We choose in (2.3) this branch of argument which is equal when
Since , we have from (2.3) where is given by (2.1). The constant cannot be smaller. Let be fixed. Let us consider the point with and Let be such that is finite. Then, let where Now, for , and Since maps the unit circle onto imaginary axis, we may choose , such that ,, This means that is finite and Hence Thus, from (2.4) and (2.6), we have Therefore cannot be smaller.
For consider the sequence , , , such that So
Let with finite and The function defined as belongs to Thus, from (2.9), it follows that This means that is best possible.
We note that, for , we obtain a result proved in [8].

Theorem 2.2. Let Then, for every and , with , one has where

Proof. To prove this result, we shall essentially use the similar method given by Kaplan [9].
Let for fixed Then satisfies the inequality (1.4) for some , and Let , Since , for we can define, for , is a real number, the following: The functions , , , and are continuous and periodic with period . From (1.4), we can choose the branches of argument of and as Now, for it is known [10] that, for ,, From (2.16), (2.17), and (2.19), we have Moreover, by (2.16), we have and therefore, from (2.20) where and are defined by (2.13). This completes the proof.

We note that, for , , we obtain the necessary condition for to be close-to-convex in proved in [9].

Remark 2.3. From Theorem 2.2, we can interpret some geometrical meaning for the functions in For simplicity, let us suppose that the image domain is bounded by an analytic curve . At a point on , the outward drawn normal turns back at most where is given by (2.13). This is a necessary condition for a function to belong to . Goodman [4] showed that if , then, for , ,

We note that is univalent for since The functions in need not even be finitely valent in for

Remark 2.4. From Theorem 2.2 and [11, Lemma  1.3] by Pommerenke, it follows that is a linearly invariant family of order Therefore, the image of under functions in contains the schlicht disk

Theorem 2.5. Let ,, , be of the form (1.1). Then This estimate is best possible, extremal function being defined by (2.4).

Proof. Let , in (1.5).
Now, it is known that, for functions of positive real part with , is subordinate to Also , see [1, 12]. Therefore, from (1.5), we have and this gives us the required result.

Remark 2.6. Let for and be given by (1.1). Then is univalent in by Remark 2.3 and is univalent in for , Now and therefore and so on using Theorem 2.5. Hence it follows that the image of under with contains the schlicht disc .

From Remark 2.3, and the results proved for the class , in [4], we at once have the following.

Theorem 2.7. Let and be given by (1.1). Let be defined by where , and is given by (2.13). Clearly (i)Denote by the length of the image of the circle under and by the area of Then, for (a)(b)(ii)For , , The function defined by (2.25), shows that this upper bound is sharp.

Theorem 2.8. Let Then, for , , , where is a constant depending upon , and only.

Proof. With , For it is known [10] that there exist such that Also, for we have for , (see [13]). Now, from (2.27), (2.28), and (2.29), we have where we have used distortion theorems, subordination for the starlike functions, and Holder’s inequality, and and are constants.

Theorem 2.9. Let and be given by (1.1). Then, for , , one has , where is a constant depending only on , , and

Proof. With Cauchy’s theorem gives Thus Using Theorem 2.8 and putting we prove this result.


The authors would like to thank the referee for thoughtful comments and suggestions.