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Directional Convexity of Harmonic Mappings

Directional Convexity of Harmonic Mappings The convolution properties are discussed for the complex-valued harmonic functions on the unit disk $$\mathbb {D}$$ D constructed from the harmonic shearing of the analytic function $$\phi (z):=\int _0^z (1-2\xi \textit{e}^{\textit{i}\mu }\cos \nu +\xi ^2\textit{e}^{2\textit{i}\mu })^{-1}{} \textit{d}\xi $$ ϕ ( z ) : = ∫ 0 z ( 1 - 2 ξ e i μ cos ν + ξ 2 e 2 i μ ) - 1 d ξ , where $$\mu $$ μ and $$\nu $$ ν are real numbers. For any real number $$\alpha $$ α and a harmonic function $$f=h+\overline{g}$$ f = h + g ¯ , define an analytic function $$f_{\alpha }$$ f α by $$f_{\alpha }:=h+\textit{e}^{-2\textit{i}\alpha }g$$ f α : = h + e - 2 i α g . Let $$\mu _1$$ μ 1 and $$\mu _2$$ μ 2 $$(\mu _1+\mu _2=\mu )$$ ( μ 1 + μ 2 = μ ) be real numbers, and $$f=h+\overline{g}$$ f = h + g ¯ and $$F=H+\overline{G}$$ F = H + G ¯ be locally univalent and sense-preserving harmonic functions such that $$f_{\mu _1}*F_{\mu _2}=\phi $$ f μ 1 ∗ F μ 2 = ϕ . It is shown that the convolution $$f*F$$ f ∗ F is univalent and convex in the direction of $$-\,\mu $$ - μ , provided it is locally univalent and sense-preserving. Also, local univalence of the above convolution $$f*F$$ f ∗ F is shown when f and F have specific analytic dilatations. Furthermore, if $$g\equiv 0$$ g ≡ 0 and both the analytic functions $$f_{\mu _1}$$ f μ 1 and $$F_{\mu _2}$$ F μ 2 are convex, then the convolution $$f*F$$ f ∗ F is shown to be convex. These results extend the work of Dorff et al. (Complex Var Elliptic Equ 57(5):489–503, 2012) to a larger class of functions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Malaysian Mathematical Sciences Society Springer Journals

Directional Convexity of Harmonic Mappings

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References (10)

Publisher
Springer Journals
Copyright
Copyright © 2017 by Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia
Subject
Mathematics; Mathematics, general; Applications of Mathematics
ISSN
0126-6705
eISSN
2180-4206
DOI
10.1007/s40840-017-0552-2
Publisher site
See Article on Publisher Site

Abstract

The convolution properties are discussed for the complex-valued harmonic functions on the unit disk $$\mathbb {D}$$ D constructed from the harmonic shearing of the analytic function $$\phi (z):=\int _0^z (1-2\xi \textit{e}^{\textit{i}\mu }\cos \nu +\xi ^2\textit{e}^{2\textit{i}\mu })^{-1}{} \textit{d}\xi $$ ϕ ( z ) : = ∫ 0 z ( 1 - 2 ξ e i μ cos ν + ξ 2 e 2 i μ ) - 1 d ξ , where $$\mu $$ μ and $$\nu $$ ν are real numbers. For any real number $$\alpha $$ α and a harmonic function $$f=h+\overline{g}$$ f = h + g ¯ , define an analytic function $$f_{\alpha }$$ f α by $$f_{\alpha }:=h+\textit{e}^{-2\textit{i}\alpha }g$$ f α : = h + e - 2 i α g . Let $$\mu _1$$ μ 1 and $$\mu _2$$ μ 2 $$(\mu _1+\mu _2=\mu )$$ ( μ 1 + μ 2 = μ ) be real numbers, and $$f=h+\overline{g}$$ f = h + g ¯ and $$F=H+\overline{G}$$ F = H + G ¯ be locally univalent and sense-preserving harmonic functions such that $$f_{\mu _1}*F_{\mu _2}=\phi $$ f μ 1 ∗ F μ 2 = ϕ . It is shown that the convolution $$f*F$$ f ∗ F is univalent and convex in the direction of $$-\,\mu $$ - μ , provided it is locally univalent and sense-preserving. Also, local univalence of the above convolution $$f*F$$ f ∗ F is shown when f and F have specific analytic dilatations. Furthermore, if $$g\equiv 0$$ g ≡ 0 and both the analytic functions $$f_{\mu _1}$$ f μ 1 and $$F_{\mu _2}$$ F μ 2 are convex, then the convolution $$f*F$$ f ∗ F is shown to be convex. These results extend the work of Dorff et al. (Complex Var Elliptic Equ 57(5):489–503, 2012) to a larger class of functions.

Journal

Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals

Published: Oct 22, 2017

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