arXiv:1501.03197v1 [math.CV] 13 Jan 2015 THE LOWER BOUND FOR THE MODULUS OF THE DERIVATIVES AND JACOBIAN OF HARMONIC UNIVALENT MAPPINGS MIODRAG MATELJEVI ´ C Research partially supported by MNTRS, Serbia, Grant No. 174 032 This is a very rough version. Abstract. We give the lower bound for the modulus of the radial derivatives and Jacobian of harmonic univalent mappings from the unit ball onto convex domain in plane and space. As an application we show that if, in addition, f is univalent harmonic K-qc gradient mapping, then f is co-Lipschitz. We also review related results in planar case using some novelity. Throughout the paper we denote by Ω, G and D open subset of R n , n ≥ 1. Let B n (x, r)= {z ∈ R n : |z − x| <r},S n−1 (x, r)= ∂B n (x, r) (abbreviated S(x, r)) and let B n , S = S n−1 stand for the unit ball and the unit sphere in R n , respectively. In particular, by D we denote the unit disc B 2 and T = ∂ D we denote the unit circle S 1 in the complex plane. For a domain D in R n with non-empty boundary, we define the distance function d = d(D) = dist(D) by d(x)= d(x; ∂D) = dist(D)(x) = inf {|x − y| : y ∈ ∂D}; and if f maps D onto D ′ ⊂ R n , in some settings it is convenient to use short notation d ∗ = d f (x) for d(f (x); ∂D ′ ). It is clear that d(x) = dist(x, D c ), where D c is the complement of D in R n . Let G be an open set in R n . A mapping f : G → R m is differentiable at x ∈ G if there is a linear mapping f ′ (x): R n → R n , called the derivative of f at x, such that f (x + h) − f (x)= f ′ (x)h + |h|ε(x, h) where ε(x, h) → 0 as h → 0. For a vector-valued function f : G → R n , where G ⊂ R n , is a domain, we define |f ′ (x)| = max |h|=1 |f ′ (x)h| and l(f ′ (x)) = min |h|=1 |f ′ (x)h| , when f is differentiable at x ∈ G. For x ∈ R n , we use notation r = |x|. We say that Jacobian J of mapping on a domain Ω satisfies minimum principle if for every compact F ⊂ Ω we have inf F J ≥ inf ∂F J . A C 1 (in particular diffemorphisam) mapping f :Ω → Ω ∗ is K-qc iff (0.1) |f ′ (x)| n /K ≤|J (x, f )|≤ Kℓ((f ′ (x)) n Date : September 10, 2014. 1