TY - JOUR

T1 - Stable geometric properties of analytic and harmonic functions

AU - Hernández, Rodrigo

AU - Martín, María J.

PY - 2013/9

Y1 - 2013/9

N2 - Given any sense preserving harmonic mapping f=h+g in the unit disk, we prove that for all |λ|=1 the functions fλ=h+λ 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.

AB - Given any sense preserving harmonic mapping f=h+g in the unit disk, we prove that for all |λ|=1 the functions fλ=h+λ 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.

UR - http://www.scopus.com/inward/record.url?scp=84880986513&partnerID=8YFLogxK

U2 - 10.1017/S0305004113000340

DO - 10.1017/S0305004113000340

M3 - Article

AN - SCOPUS:84880986513

VL - 155

SP - 343

EP - 359

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 2

ER -