Contractions of harmonic univalent functions (Q2747349)

For \(0 < c \leq 1\), \(0\leq \alpha < 1\), let \(H_cS^*(\alpha)\) be the subclass of starlike harmonic functions of order \(\alpha\), consisting of functions \(f=g+\overline{h}\), \(g(z)=z+\sum_2^\infty a_nz^n\), \(h(z)=\sum_1^\infty b_nz^n\) satisfying \(\sum_2^\infty(n-\alpha)|a_n|+\sum_1^\infty(n+\alpha)|b_n|\leq 1-\alpha\). The subclass \(H_cK(\alpha)\) consisting of convex harmonic functions is defined similarly. Let \(H_c^0S^*(\alpha)\) and \(H_c^0K(\alpha)\) be the corresponding classes when \(b_1=0\). The authors have determined a sharp upper bound of \(c\) for the inclusion \(H_c^0S^*(\alpha) \subseteq H^0K(\beta)\). They obtained extreme points, sharp coefficient and distortion bounds for the classes \(H_c^0S^*(\alpha)\) and \(H_c^0K(\alpha)\). Also they have found \(\beta=\beta(\alpha,c)\) such that \(H_c^0S^*(\alpha) \subset H_c^0S^*(\beta)\). The distortion bounds for \(H_cS^*(\alpha)\) and \(H_cK(\alpha)\) are obtained.