Application of the Subordination Principle to the Harmonic Mappings Convex in One Direction with Shear Construction Method
© Yaşar Polatoğlu et al. 2010
Received: 3 June 2010
Accepted: 26 July 2010
Published: 11 August 2010
Any harmonic function in the open unit disc can be written as a sum of an analytic and antianalytic functions , where and are analytic functions in and are called the analytic part and the coanalytic part of respectively. Many important questions in the study of the classes of functions are related to bounds on the modulus of functions (growth) or the modulus of the derivative (distortion). In this paper, we consider both of these questions.
Let be a simply connected domain in the complex plane. A harmonic function has the representation , where and are analytic in and are called the analytic and coanalytic parts of , respectively. Let , and be analytic functions in the open unit disc . If , then is called the sense-preserving harmonic univalent function in . The class of all sense-preserving harmonic univalent functions is denoted by , with , , and , and the class of all sense-preserving harmonic univalent functions is denoted by with , . For convenience, we will examine sense-preserving functions, that is, functions for which . If has , then is sense preserving. The analytic dilatation of the harmonic functions is given by . We also note that if is locally univalent and sense preserving then .
In this paper we examine the class of functions that are convex in one direction. The shear construction is essential to the present work as it allows one to study harmonic functions through their related analytic functions as shown in  by Hengartner and Schober. The shear construction produces a univalent harmonic function that maps to the region that is convex in the direction of the real axis. This construction relies on the following theorem of Clunie and Sheil-Small.
Theorem 1.1 (see ).
A harmonic function locally univalent in is a univalent mapping of onto a domain convex in the direction of the real axis if and only if is a conformal univalent mapping of onto a domain convex in the direction of the real axis.
Theorem 1.2 (see ).
Using this characterization of functions, Hengartner and Schober proved the following theorem.
Theorem 1.3 (see ).
To be able to obtain this result for functions that are in the direction of the real axis, let us consider the following situation. Suppose that is a function that is analytic and convex in the direction of the real axis. Furthermore, suppose that is normalized by the following.
Consequently, if satisfies (1.1), then satisfies (1.3). Knowing this, we can apply and see that the result still holds, with being replaced by . In this situation, . We can now prove the derivative bounds for the harmonic function convex in the direction of the real axis.
Finally, let be the family of functions which are analytic in and satisfying the condition , for every . Denote by the class of analytic functions given by which satisfy for all . Let and be analytic functions in . If is satisfied for some and every , then we say that is subordinate to , and we write .
2. Main Results
Following some simple calculations from (2.8), we get (2.1), (2.2), (2.3), and (2.4).
These distortions were found by Schaubroeck .
Applying (2.9) to the above expression yields (2.14).
This growth was found by Schaubroeck .
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