Journal
MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY
Volume 155, Issue 2, Pages 343-359Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1017/S0305004113000340
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Funding
- Fondecyt, Chile [1110160, 1110321]
- MICINN [MTM2009-14694-C02-01]
- MINECO, Spain [MTM2012-37436-C02-02]
- Fondecyt [1110160]
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Given any sense preserving harmonic mapping f = h + (g) over bar in the unit disk, we prove that for all vertical bar lambda vertical bar = 1 the functions f(lambda) = h + lambda(g) over bar are univalent (resp. close-to-convex, starlike, or convex) if and only if the analytic functions F-lambda = h + lambda g are univalent (resp. close-to-convex, starlike, or convex) for all such lambda. We also obtain certain necessary geometric conditions on h in order that the functions f(lambda) belong to the families mentioned above. In particular, we see that if f(lambda) are univalent for all lambda on the unit circle, then h is univalent.
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