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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 820696, 10 pages
A Note on the Class of Functions with Bounded Turning
1Institute of Mathematical Sciences, University Malaya, 50603 Kuala Lumpur, Malaysia
2Department 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.
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.
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 ). 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 , Mocanu obtained a subclass of , which is contained in . Recently, Tuneski generalized the class of convex functions with bounded turning (see ): 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 ). 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 .
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:
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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