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Stable geometric properties of analytic and harmonic functions

Published online by Cambridge University Press:  04 June 2013

RODRIGO HERNÁNDEZ  and
MARÍA J. MARTÍN
RODRIGO HERNÁNDEZ
Affiliation:
Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av. Padre Hurtado 750, Viña del Mar, Chile. e-mail: rodrigo.hernandez@uai.cl
MARÍA J. MARTÍN
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, Módulo 17 (Edificio de Ciencias), 28049 Madrid, Spain. e-mail: mariaj.martin@uam.es URL: http://www.uam.es/mariaj.martin
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Abstract

Given any sense preserving harmonic mapping f=h+ḡ in the unit disk, we prove that for all |λ|=1 the functions fλ=h+λḡ are univalent (resp. close-to-convex, starlike, or convex) if and only if the analytic functions Fλ=h+λg are univalent (resp. close-to-convex, starlike, or convex) for all such λ. We also obtain certain necessary geometric conditions on h in order that the functions fλ belong to the families mentioned above. In particular, we see that if fλ are univalent for all λ on the unit circle, then h is univalent.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 2 , September 2013 , pp. 343 - 359
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

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