[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.2014.51.3.803

HARMONIC MAPPINGS RELATED TO FUNCTIONS WITH BOUNDED BOUNDARY ROTATION AND NORM OF THE PRE-SCHWARZIAN DERIVATIVE

Kanas, Stanis lawa (Stanis lawa Kanas University of Rzeszow)
Klimek-Smet, Dominika (Maria Curie-Sk lodowska University)

Publication Information

Bulletin of the Korean Mathematical Society / v.51, no.3, 2014 , pp. 803-812 More about this Journal

Abstract

Let ${\mathcal{S}}^0_{\mathcal{H}}$ be the class of normalized univalent harmonic mappings in the unit disk. A subclass ${\mathcal{V}}^{\mathcal{H}}(k)$ of ${\mathcal{S}}^0_{\mathcal{H}}$ , whose analytic part is function with bounded boundary rotation, is introduced. Some bounds for functionals, specially harmonic pre-Schwarzian derivative, described in ${\mathcal{V}}^{\mathcal{H}}(k)$ are given.

Keywords

univalent harmonic mappings; functions with bounded boundary rotation;

Citations & Related Records

Reference

1	A. J. H. Vincent, Sur la resolution des equations numeriques, J. Math. Pures Appl. 1 (1836), 341-372.
2	F. G. Avkhadiev and K. J. Wirths, Schwarz-Pick Type Inequalities, Birkhauser Verlag AG, Basel-Boston-Berlin, 2009.
3	J. G. Clunie and T. Sheil-Small, Harmonic Univalent Functions, Ann. Acad. Sci. Fenn. Ser. Math. 9 (1984), 3-25. DOI
4	F. Colonna, The Bloch constant of bounded harmonic mappings, Indiana Univ. Math. J. 38 (1989), no. 4, 829-840. DOI
5	M. Chuaqui, P. Duren, and B. Osgood, The Schwarzian derivative for harmonic mappings, J. Anal. Math. 91 (2003), 329-351. DOI
6	P. L. Duren and O. Lehto, Schwarzian derivatives and homeomorphic extensions, Ann. Acad. Sci. Fenn. Ser. 477 (1970), 11 pp.
7	J. Fourier, Analyse des equations determinees, F. Didot, Paris, 1831.
8	A. W. Goodman, Univalent Functions, Mariner Publishing, Tampa, 1983.
9	D. Kalaj, S. Ponnusamy, and M. Vuorinen, Radius of close-to-convexity of harmonic functions, Complex Var. Elliptic Equ. (2013), to appear.(arXiv:1107.0610)
10	S. Kanas and D. Klimek, Coefficient estimates and Bloch's constant in some class of harmonic mappings, to appear.
11	D. Klimek and A. Michalski, Univalent anti-analytic perturbations of the identity in the unit disc, Sci. Bull. Che lm 1 (2006), 67-76.
12	D. Klimek and A. Michalski, Univalent anti-analytic perturbations of convex analytic mappings in the unit disc, Ann. Univ. Mariae Curie-Sk lodowska Sect. A Vol. LXI (2007), Sectio A, 39-49.
13	W. Krauss, Uber den Zusammenhang einiger Clarakteristiken eines einfach zusammenh angenden Bereiches mit der Kreisabbildung, Mitt. Math. Sem. Giessen 21 (1932), 1-28.
14	O. Lehto, On the distortion of conformal mappings with bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. Math. 124 (1952), no. 124, 14pp.
15	H. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), no. 10, 689-692. DOI
16	Z. Nehari, The Schwarzian derivatives and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545-551. DOI
17	L. V. Ahlfors and G. Weill, A uniqueness theorem for Beltrami equations, Proc. Amer. Math. Soc. 13 (1962), 975-978. DOI ScienceOn
18	P. L. Duren, Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics 156, Cambridge Univ. Press, Cambridge, 2004.
19	A. Alesina and M. Galuzzi, Vincent's theorem from a modern point of view, In R. Betti and W. F. Lawvere (Eds.), Categorical Studies in Italy 2000, Rend. Circ. Mat. Palermo, Serie II 64 (2000), 179-191.
20	A. G. Akritas, A. W. Strzebonski, and P. S. Vigklas, Improving the performance of the continued fractions method using new bounds of positive roots, Nonlinear Anal. Model. Control 13 (2008), no. 3, 265-279.