Coefficients Estimate for Harmonic v-Bloch Mappings and Harmonic K-Quasiconformal Mappings

• Published in: Bulletin of the Malaysian Mathematical Sciences Society 2016, Vol.39 (1), p.349-358
• Main Author: Zhu, Jian-Feng
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Mapping; Mathematics; Mathematics and Statistics; Theorems
• ISSN: 0126-6705; 2180-4206
• DOI: 10.1007/s40840-015-0175-4

Let f ( z ) = h ( z ) + g ( z ) ¯ be a harmonic v -Bloch mapping defined in the unit disk D with ‖ f ‖ B v ≤ M , where h ( z ) = ∑ n = 1 ∞ a n z n and g ( z ) = ∑ n = 1 ∞ b n z n are analytic in D . In this paper, we obtain the coefficient estimates for f as follows: | a n | 2 + | b n | 2 ≤ A n ( v , M ) , where A n ( v , M ) is given in Theorem 1 . Furthermore, we prove that for v < 1 , lim n → ∞ A n ( v , M ) = 0 and for v ≥ 1 , A n ( v , M ) ≤ O ( n 2 v - 2 ) . Moreover, if f is a harmonic K -quasiconformal self-mapping of D , then | a n | + | b n | ≤ B n ( K ) , where B n ( K ) is given in Theorem 3 such that lim n → ∞ B n ( K ) = 0 and B n ( 1 ) = 4 n π .