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Application of the subordination principle to the harmonic mappings convex in one direction with shear construction method. (English) Zbl 1200.30022

Summary: Any harmonic function in the open unit disc \(\mathbb D=\{ z\,|\,| z|<1\}\) can be written as a sum of an analytic and an antianalytic function \(f(z)=h(z)+\overline{g(z)}\), where \(h(z)\) and \(g(z)\) are analytic functions in \(\mathbb D\), and are called the analytic part and the coanalytic part of \(f\), respectively. Many important questions in the study of 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.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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[1] Hengartner W, Schober G: On Schlicht mappings to domains convex in one direction. Commentarii Mathematici Helvetici 1970, 45: 303-314. 10.1007/BF02567334 · Zbl 0203.07604 · doi:10.1007/BF02567334
[2] Clunie J, Sheil-Small T: Harmonic univalent functions. Annales Academiae Scientiarum Fennicae. Series A I. Mathematica 1984, 9: 3-25. · Zbl 0506.30007 · doi:10.5186/aasfm.1984.0905
[3] Schaubroeck LE: Growth, distortion and coefficient bounds for plane harmonic mappings convex in one direction. The Rocky Mountain Journal of Mathematics 2001, 31(2):625-639. 10.1216/rmjm/1020171580 · Zbl 0995.30009 · doi:10.1216/rmjm/1020171580
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