On harmonic entire mappings II

In this paper, we investigate properties of harmonic entire mappings. First, we study lower order of harmonic entire mappings. For a harmonic entire mappings f of order ρ , we also discuss the case ρ= ∞ by introducing the quantities ρ(k) , τ(k) , λ(k) , ω(k) , and also the case ρ= 0 by studying logarithmic order ρl , logarithmic type τl , logarithmic lower order λl , and logarithmic lower type ωl . Secondly, we investigate approximation by harmonic polynomials of harmonic entire mappings. For a real valued continuous function f on [ - 1 , 1 ] , let En(f)=infpn∈πn‖f-pn‖,n=0,1,2,⋯, where the norm is the maximum norm on [ - 1 , 1 ] and πn denotes the class of all harmonic polynomials with real coefficients of degree at most n. It is known that limn→∞En1/n(f)=0 if and only if f is the restriction to [ - 1 , 1 ] of an entire function (cf. [5, Theorem 7, p. 76]). We prove that this result continues to hold for harmonic entire mappings. We also study the relationship of ρ(k) and λ(k) with the rate growth of En1/n(f) and investigate the relationship of ρl , τl , λl , ωl with the asymptotic behaviour of En1/n(f) .