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

Title & Authors
HARMONIC MAPPINGS RELATED TO FUNCTIONS WITH BOUNDED BOUNDARY ROTATION AND NORM OF THE PRE-SCHWARZIAN DERIVATIVE
Kanas, Stanis lawa; Klimek-Smet, Dominika;

Abstract
Let $\small{{\mathcal{S}}^0_{\mathcal{H}}}$ be the class of normalized univalent harmonic mappings in the unit disk. A subclass $\small{{\mathcal{V}}^{\mathcal{H}}(k)}$ of $\small{{\mathcal{S}}^0_{\mathcal{H}}}$, whose analytic part is function with bounded boundary rotation, is introduced. Some bounds for functionals, specially harmonic pre-Schwarzian derivative, described in $\small{{\mathcal{V}}^{\mathcal{H}}(k)}$ are given.
Keywords
univalent harmonic mappings;functions with bounded boundary rotation;
Language
English
Cited by
References
1.
L. V. Ahlfors and G. Weill, A uniqueness theorem for Beltrami equations, Proc. Amer. Math. Soc. 13 (1962), 975-978.

2.
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.

3.
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.

4.
F. G. Avkhadiev and K. J. Wirths, Schwarz-Pick Type Inequalities, Birkhauser Verlag AG, Basel-Boston-Berlin, 2009.

5.
M. Chuaqui, P. Duren, and B. Osgood, The Schwarzian derivative for harmonic mappings, J. Anal. Math. 91 (2003), 329-351.

6.
J. G. Clunie and T. Sheil-Small, Harmonic Univalent Functions, Ann. Acad. Sci. Fenn. Ser. Math. 9 (1984), 3-25.

7.
F. Colonna, The Bloch constant of bounded harmonic mappings, Indiana Univ. Math. J. 38 (1989), no. 4, 829-840.

8.
P. L. Duren, Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics 156, Cambridge Univ. Press, Cambridge, 2004.

9.
P. L. Duren and O. Lehto, Schwarzian derivatives and homeomorphic extensions, Ann. Acad. Sci. Fenn. Ser. 477 (1970), 11 pp.

10.
J. Fourier, Analyse des equations determinees, F. Didot, Paris, 1831.

11.
A. W. Goodman, Univalent Functions, Mariner Publishing, Tampa, 1983.

12.
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)

13.
S. Kanas and D. Klimek, Coefficient estimates and Bloch's constant in some class of harmonic mappings, to appear.

14.
D. Klimek and A. Michalski, Univalent anti-analytic perturbations of the identity in the unit disc, Sci. Bull. Che lm 1 (2006), 67-76.

15.
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.

16.
W. Krauss, Uber den Zusammenhang einiger Clarakteristiken eines einfach zusammenh angenden Bereiches mit der Kreisabbildung, Mitt. Math. Sem. Giessen 21 (1932), 1-28.

17.
O. Lehto, On the distortion of conformal mappings with bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. Math. 124 (1952), no. 124, 14pp.

18.
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.

19.
Z. Nehari, The Schwarzian derivatives and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545-551.

20.
A. J. H. Vincent, Sur la resolution des equations numeriques, J. Math. Pures Appl. 1 (1836), 341-372.