Regularity theorems for harmonic functions

Abstract

The regularity theorem is a result stating that functions which have extremal growth or decrease in the given class display a regular behaviour. Such theorems for linearly invariant families of analytic functions are well known. We prove regularity theorems for some classes of harmonic functions. Many presented statements are new even in the analytic case.

Coefficients of univalent harmonic mappings

“…Let us compute the approximate value of r n,n for large values of n. By setting r = 1−x/n in k(n, r), and making use of the fact that (1 − x/n) n ≤ e −x for x ≥ 0, we get that k(n, 1 − x/n) ≤ t(x, n), where t(x, n) := 177e −x n 8 50x 8 2 − x n 9 q(x, n) 16n 4 − 32n 3 x + 28n 2 x 2 − 12nx 3 + 3x 4 , with q(x, n) = n[n 3 (24 + 24x + 12x 2 + 4x 3 + x 4 ) − 6n 2 x(6 + 4x + x 2 ) + 2nx 2 (7 + 2x) − x 3 ].…”

Coefficients of univalent harmonic mappings

“…Let f = h + g ∈ U 0 H (16.5). From the power series representation of h(z) given by (1) and Lemma E, we obtain that (8) |a 15 (1 − r) 18 =: ψ n (r), where 0 < r < 1. In particular,…”

On Regularity Theorems for Linearly Invariant Families of Harmonic Functions

“…Denote R 1 (r)e iγ 1 (r) = re iϕ 1 + a 1 +āre iϕ 1 , where γ 1 (r) is a real-valued function. Then, using (11) for R(r) = R 1 (r), we obtain…”

“…It was shown in [10] that ord U H α ≥ 1. In [11] and [12], the following regularity theorems for harmonic functions were proved:…”