ON HARMONIC CONVOLUTIONS INVOLVING A VERTICAL STRIP MAPPING

Title & Authors
ON HARMONIC CONVOLUTIONS INVOLVING A VERTICAL STRIP MAPPING
Kumar, Raj; Gupta, Sushma; Singh, Sukhjit; Dorff, Michael;

Abstract
Let $\small{f_{\beta}=h_{\beta}+\bar{g}_{\beta}}$ and $\small{F_a=H_a+\bar{G}_a}$ be harmonic mappings obtained by shearing of analytic mappings $\small{h_{\beta}+g_{\beta}=1/(2isin{\beta})log$$(1+ze^{i{\beta}})/(1+ze^{-i{\beta}})$$}$, 0 < $\small{{\beta}}$ < $\small{{\pi}}$ and $\small{H_a+G_a=z/(1-z)}$, respectively. Kumar et al. [7] conjectured that if $\small{{\omega}(z)=e^{i{\theta}}z^n({\theta}{\in}\mathbb{R},n{\in}\mathbb{N})}$ and $\small{{\omega}_a(z)=(a-z)/(1-az)}$, $\small{a{\in}(-1,1)}$ are dilatations of $\small{f_{\beta}}$ and $\small{F_a}$, respectively, then $\small{F_a\tilde{\ast}f_{\beta}{\in}S^0_H}$ and is convex in the direction of the real axis, provided $\small{a{\in}[(n-2)/(n+2),1)}$. They claimed to have verified the result for n = 1, 2, 3 and 4 only. In the present paper, we settle the above conjecture, in the affirmative, for $\small{{\beta}={\pi}/2}$ and for all $\small{n{\in}\mathbb{N}}$.
Keywords
univalent harmonic mapping;vertical strip mapping;harmonic convolution;
Language
English
Cited by
1.
A NOTE ON CONVEXITY OF CONVOLUTIONS OF HARMONIC MAPPINGS,;;;

대한수학회보, 2015. vol.52. 6, pp.1925-1935
1.
On harmonic K-quasiconformal mappings associated with asymmetric vertical strips, Acta Mathematica Sinica, English Series, 2015, 31, 12, 1970
2.
A NOTE ON CONVEXITY OF CONVOLUTIONS OF HARMONIC MAPPINGS, Bulletin of the Korean Mathematical Society, 2015, 52, 6, 1925
References
1.
J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I Math. 9 (1984), 3-25.

2.
M. Dorff, Harmonic univalent mappings onto asymmetric vertical strips, Computational methods and function theory 1997 (Nicosia), 171-175, Ser. Approx. Decompos., 11, World Sci. Publ., River Edge, NJ, 1999.

3.
M. Dorff, Convolutions of planar harmonic convex mappings, Complex Var. Theory Appl. 45 (2001), no. 3, 263-271.

4.
M. Dorff, M. Nowak, and M.Woloszkiewicz, Convolutions of harmonic convex mappings, Complex Var. Elliptic Equ. 57 (2012), no. 5, 489-503.

5.
W. Hengartner and G. Schober, Univalent harmonic functions, Trans. Amer. Math. Soc. 299 (1987), no. 1, 1-31.

6.
R. Kumar, M. Dorff, S. Gupta, and S. Singh, Convolution properties of some harmonic mappings in the right half-plane, see arXiv:1206.4364.

7.
R. Kumar, S. Gupta, S. Singh, and M. Dorff, An application of Cohn's rule to convolutions of univalent harmonic mappings, see arXiv:1306.5375.

8.
L. Li and S. Ponnusamy, Convolutions of slanted half-plane harmonic mappings, Analysis (Munich) 33 (2013), no. 2, 159-176.

9.
L. Li and S. Ponnusamy, Sections of stable harmonic convex functions, Nonlinear Analysis 2014 (2014), 11 pages; http:/dx.doi.org/10.1016/j.na.2014.06.005.

10.
L. Li and S. Ponnusamy, Convolutions of harmonic mappings convex in one direction, Complex Anal. Oper. Th. 2014 (2014), 17 pages (available online: DOI 10.1007/s11785-014-0394-y).

11.
L. Li, S. Ponnusamy, and M. Vuorinen, The minimal surfaces over the slanted halfplanes, vertical strip and singlr slit, See http://arxiv.org/pdf/1204.2890.pdf

12.
Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, London Mathematical Society Monographs New Series, Vol. 26, Oxford University Press, Oxford, 2002.