Coefficient estimates and radii problems for certain classes of polyharmonic mappings

Abstract

We give coefficient estimates for a class of close-to-convex harmonic mappings , and discuss the Fekete–Szegő problem of it. We also determine a disk in which the partial sum is close-to-convex for each . Then, we introduce two classes of polyharmonic mappings and , consider the starlikeness and convexity of them and obtain coefficient estimates for them. Finally, we give a necessary condition for a mapping to be in the class .

Keywords:

• harmonic mapping
• polyharmonic mapping
• coefficient estimates
• the Fekete–Szegő problem
• partial sum
• close-to-convex
• starlike
• convex

AMS Subject Classifications:

• 30C45
• 30C50

Acknowledgements

The authors are indebted to the referees for their helpful remarks.

Notes

1 The research was partly supported by Hunan Provincial Innovation Foundation for Postgraduates [number 125000-4242]; CIMO Scholarship of Finland and China Scholarship Council, Academy of Finland [number 269260] and National Natural Science Foundation of China.