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Convex Combinations of Planar Harmonic Mappings Realized Through Convolutions with Half-Strip Mappings

Convex Combinations of Planar Harmonic Mappings Realized Through Convolutions with Half-Strip... Recent investigations into what geometric properties are preserved under the convolution of two planar harmonic mappings on the open unit disk $${\mathbb {D}}$$ D have typically involved half-plane and strip mappings. These results rely on having a convolution that is locally univalent and sense-preserving on $${\mathbb {D}}$$ D , and thus, much focus has been on trying to satisfy this condition. We introduce a family of right half-strip harmonic mappings, $$\Psi _c : {\mathbb {D}}\rightarrow {\mathbb {C}}$$ Ψ c : D → C , $$c>0$$ c > 0 , and consider the convolution $$\Psi _c * f$$ Ψ c ∗ f for a harmonic mapping $$f = h +\overline{g}: {\mathbb {D}}\rightarrow {\mathbb {C}}$$ f = h + g ¯ : D → C . We prove it is sufficient for $$h \pm g$$ h ± g to be starlike for $$\Psi _c *f$$ Ψ c ∗ f to be locally univalent and sense-preserving. Moreover, $$\Psi _c * f$$ Ψ c ∗ f decomposes into a convex combination of two harmonic mappings, one of which is f itself. This decomposition is key in addressing mapping properties of the convolution, and from it, we produce a family of convex octagonal harmonic mappings as well some other families of convex harmonic mappings. Additionally, motivated by the construction of $$\Psi _c$$ Ψ c , we introduce a generalized harmonic Bernardi integral operator. We demonstrate convolution preserving properties and a weak subordination relationship for this extended operator. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Malaysian Mathematical Sciences Society Springer Journals

Convex Combinations of Planar Harmonic Mappings Realized Through Convolutions with Half-Strip Mappings

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References (22)

Publisher
Springer Journals
Copyright
Copyright © 2016 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/s40840-016-0336-0
Publisher site
See Article on Publisher Site

Abstract

Recent investigations into what geometric properties are preserved under the convolution of two planar harmonic mappings on the open unit disk $${\mathbb {D}}$$ D have typically involved half-plane and strip mappings. These results rely on having a convolution that is locally univalent and sense-preserving on $${\mathbb {D}}$$ D , and thus, much focus has been on trying to satisfy this condition. We introduce a family of right half-strip harmonic mappings, $$\Psi _c : {\mathbb {D}}\rightarrow {\mathbb {C}}$$ Ψ c : D → C , $$c>0$$ c > 0 , and consider the convolution $$\Psi _c * f$$ Ψ c ∗ f for a harmonic mapping $$f = h +\overline{g}: {\mathbb {D}}\rightarrow {\mathbb {C}}$$ f = h + g ¯ : D → C . We prove it is sufficient for $$h \pm g$$ h ± g to be starlike for $$\Psi _c *f$$ Ψ c ∗ f to be locally univalent and sense-preserving. Moreover, $$\Psi _c * f$$ Ψ c ∗ f decomposes into a convex combination of two harmonic mappings, one of which is f itself. This decomposition is key in addressing mapping properties of the convolution, and from it, we produce a family of convex octagonal harmonic mappings as well some other families of convex harmonic mappings. Additionally, motivated by the construction of $$\Psi _c$$ Ψ c , we introduce a generalized harmonic Bernardi integral operator. We demonstrate convolution preserving properties and a weak subordination relationship for this extended operator.

Journal

Bulletin of the Malaysian Mathematical Sciences SocietySpringer Journals

Published: Mar 11, 2016

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