Abstract and Applied Analysis

Volume 2012, Article ID 820696, 10 pages

http://dx.doi.org/10.1155/2012/820696

## A Note on the Class of Functions with Bounded Turning

^{1}Institute of Mathematical Sciences, University Malaya, 50603 Kuala Lumpur, Malaysia^{2}Department of Mathematics, University of Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Received 1 December 2011; Revised 25 January 2012; Accepted 29 January 2012

Academic Editor: M. Victoria Otero-Espinar

Copyright © 2012 Rabha W. Ibrahim and Adem Kılıçman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider subclasses of functions with bounded turning for normalized analytic functions in the unit disk. The geometric representation is introduced, some subordination relations are suggested, and the upper bound of the pre-Schwarzian norm for these functions is computed. Moreover, by employing Jack's lemma, we obtain a convex class in the class of functions of bounded turning and relations with other classes are posed.

#### 1. Introduction

Let be the unit disk in the complex plane , and let denote the space of all analytic functions on . Here, we suppose that is a topological vector space endowed with the topology of uniform convergence over compact subsets of . Also, for and , let be the subspace of consisting of functions of the form Further, let and let denote the class of univalent functions in . A function is called starlike if is a starlike domain with respect to the origin, and the class of univalent starlike functions is denoted by . It is called convex if is a convex domain. Each univalent starlike function is characterized by the analytic condition in . Also, it is known that is starlike if and only if is convex, which is characterized by the analytic condition in . Let , and let be a univalent function in , with . Then, we say that is subordinate to (or is superordinate to ), denoted by , if . For two functions , the Hadamard product is defined by where and are the coefficients of and , respectively.

The pre-Schwarzian derivative of is defined by with the norm It is known that if and only if is uniformly locally univalent. It is also known that for and that for . Moreover, it is showed that, when is univalent in and when , is starlike in (see [1]). Recently, the sharp norm estimates for well-known integral operators are determined (see [2–4]).

For , let denote the class of functions of the form (2.2) so that in . The functions in are called functions of bounded turning. By the Nashiro-Warschowski theorem, the functions in are univalent and also close to convex in . It is well known that and . In [5], Mocanu obtained a subclass of , which is contained in . Recently, Tuneski generalized the class of convex functions with bounded turning (see [6]): Different studies of the class of bounded turning functions can be found in [7–10].

In this note we pose the following subclass of bounded turning functions in the the unit disk: for given numbers and , let us consider the class : It is easy to see that if and only if For the development of the paper we need to define a set by all points on the right half-plane as follows: Thus it is easy to verify that its boundary satisfies the equation of a parabola

#### 2. Main Results

First, our result is in the following form.

Theorem 2.1. *A function , , if and only if there exists an analytic function ,
**
such that
**
Moreover, if function takes the form
**
then the subordination relation
**
holds.*

*Proof. *Let , and let
Integrating this equation yields (2.2). If is given in (2.2) with an analytic function , then by a differentiation of (2.2) we obtain that ; therefore,
and consequently .

Now we proceed to prove that . For this purpose we will show that the set
Let . Then
Multiplying these inequalities, we obtain
Therefore, .

Define a function , , as follows:
We suppose that
and thus
Hence, . Putting in (2.2) implies (2.3). To prove the subordination relation (2.4), first we show that is a convex function. We observe that
Let us consider the function
Computations give
Also, a calculation implies that
The aim is to show that has positive real part in the unit disk. For and a suitable choice of such that and by using (2.17), we have
Consequently, we obtain that , and therefore is a convex function.

Now by using the fact that if , satisfy and , then and is a convex function, immediately we establish (2.4). This completes the proof.

Next we consider another class of functions of bounded turning. We will estimate the upper bound of these functions by using the pre-Schwarzian norm.

Theorem 2.2. *Consider the class , of functions of bounded turning which satisfies the relation
**
Then,
*

*Proof. *Let , and let . Then, there exists an analytic function with and
Define a function such that , that is,
and thus
We proceed to determine the quantities and . Logarithmic differentiation of (2.22) yields that
and the logarithmic differentiation of (2.21) gives
Thus, by triangle inequality and the Schwarz-Pick lemma, we obtain
Consequently, we have
Therefore, , and this inequality is sharp. Thus, to determine the upper estimate of it is enough to compute . Letting , we have
Hence, we obtain (2.20).

By applying Jack’s lemma, we pose the sufficient conditions for convex functions belonging to the subclasses and .

Lemma 2.3 ( see [11]). * Let be analytic in with . If attains its maximum value on the circle at a point , then
**
where is a real number and .*

Theorem 2.4. *Assume that and . If is a convex function in of order , then .*

*Proof. *Let , that is,
From the proof of Theorem 2.2, we have
where is analytic in and satisfies and
Suppose that there exists a point such that
Then, on using Lemma 2.3 and letting and , yields
which contradicts hypothesis (2.30). Therefore, we conclude that for all , that is, ).

Theorem 2.5. *Assume that and . If satisfies
**
then .*

*Proof. *Define a function by
Then, is analytic in and satisfies . It follows that
In the same manner of Theorem 2.4, we find that
which contradicts hypothesis (2.35). Therefore, we conclude that for all , that is, .

Corollary 2.6. *Let the assumptions of Theorem 2.5 hold. Then, is strongly close to convex of order .*

*Proof. *Since , there exists an analytic function such that , and
But
which implies
and thus is strongly close to convex of order .

#### 3. Conclusion

It is well known that the class of bounded turning functions is not included in the class of the starlike functions and also the starlike functions cannot embed in the class of bounded turning functions. From the above, we conclude that some classes of bounded turning functions can be included in the class of convex functions (; Theorem 2.4). Moreover, some classes of bounded turning functions can embed in the class of close-to-convex functions (; Corollary 2.6). Hence, we have the following inclusion relation:

#### Acknowledgments

The authors would also like to thank the referee for valuable remarks and suggestions on the previous version of the paper. This research has been funded by University of Malaya, under Grant no. RG208-11AFR.

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