Radius of fully starlikeness and fully convexity of harmonic linear differential operator

2017/08/13 by Liu, ZhiHong, Ponnusamy, Saminathan
#30C45 #31C05 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1708.03883

Abstract

Let f=h+g be a normalized harmonic mapping in the unit disk \ID. In this paper, we obtain the sharp radius of univalence, fully starlikeness and fully convexity of the harmonic linear differential operators D_f^ε=zf_z-εzf_z~(|ε|=1) and F_λ(z)=(1-λ)f+λD_f^ε~(0≤λ≤ 1) when the coefficients of h and g satisfy harmonic Bieberbach coefficients conjecture conditions. Similar problems are also solved when the coefficients of h and g satisfy the corresponding necessary conditions of the harmonic convex function f=h+g. All results are sharp. Some of the results are motivated by the work of Kalaj et al. \citeKalaj2014 (Complex Var. Elliptic Equ. 59(4) (2014), 539--552).

Related

• Radius Properties of Harmonic Mappings with Fixed Analytic Part
• Starlikeness, convexity and close-to-convexity of harmonic mappings
• Radius of Close-to-convexity of Harmonic Functions
• Fully Starlike and Convex Harmonic Mappings of order α
• Coefficient Conditions for Harmonic Univalent Mappings and Hypergeometric Mappings