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 ρ l , logarithmic type τ l , logarithmic lower order λ l , and logarithmic lower type ω l . Secondly, we investigate approximation by harmonic polynomials of harmonic entire mappings. For a real valued continuous function f on [ - 1 , 1 ] , let E n ( f ) = inf p n ∈ π n ‖ f - p n ‖ , n = 0 , 1 , 2 , ⋯ , where the norm is the maximum norm on [ - 1 , 1 ] and π n denotes the class of all harmonic polynomials with real coefficients of degree at most n . It is known that lim n → ∞ E n 1 / 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 E n 1 / n ( f ) and investigate the relationship of ρ l , τ l , λ l , ω l with the asymptotic behaviour of E n 1 / n ( f ) .