The generalized Roper-Suffridge extension operator

By S. Gong, Tai-Shun Liu. 2003

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Abstract Let f(z) be a normalized convex (starlike) function on the unit disc D. Let Ω={z∈ C n :|z 1 | 2 +|z 2 | p 2 +⋯+|z n | p n , where z=(z1,z2,…,zn), z1∈D, (z 2 ,…,z n )∈ C n−1 , pi⩾1, i=2,…,n, are real numbers. In this note, we prove that Φ(f)(z)=(f(z1),f′(z1)1/p2z2,…,f′(z1)1/pnzn) is a normalized convex (starlike) mapping on Ω , where we choose the power function such that (f′(z1))1/pi|z1=0=1, i=2,…,n. Some other related results are proved.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d225a1c71e4b496c60c4ea77ea11c3cedb139108