Landau-type theorems and bi-Lipschitz theorems for bounded biharmonic mappings
- PDF / 417,926 Bytes
- 24 Pages / 439.37 x 666.142 pts Page_size
- 48 Downloads / 153 Views
Landau-type theorems and bi-Lipschitz theorems for bounded biharmonic mappings Shi-Fei Chen1 · Ming-Sheng Liu1 Received: 26 May 2020 / Accepted: 18 June 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract In this paper, we first establish five versions of Landau-type theorems for five classes of bounded biharmonic mappings F(z) = |z|2 G(z) + H (z) on the unit disk D with G(0) = H (0) = J F (0) − 1 = 0, which improve the related results of earlier authors. In particular, two versions of those Landau-type theorems are sharp. Then we derive five bi-Lipschitz theorems for these classes of bounded and normalized biharmonic mappings. Keywords Biharmonic mappings · Harmonic mappings · Landau-type theorems · Bi-Lipschitz theorems · Univalent Mathematics Subject Classification Primary 30C99; Secondary 30C62
1 Introduction Let D={z ∈ C : |z| < 1} denote the unit disk with center at the origin and radius 1. For r > 0, let Dr = {z ∈ C : |z| < r }. A function f (z) = u(z) + iv(z), z = x + i y is a harmonic mapping on the unit disk D if and only if F is twice continuously differentiable and satisfies the Laplacian equation f = 4 f z z¯ =
∂2 f ∂2 f + =0 ∂x2 ∂ y2
Communicated by Adrian Constantin.
B
Ming-Sheng Liu [email protected] Shi-Fei Chen [email protected]
1
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, People’s Republic of China
123
S.-F. Chen, M.-S. Liu
for z ∈ D, where the formal derivatives of f are defined by fz =
1 fx − i f y , 2
f z¯ =
1 fx + i f y . 2
A function F(z) = U (z) + i V (z) is a biharmonic mapping on D if and only if F is four times continuously differentiable and satisfies the biharmonic equation (F) = 0 for z ∈ D. In other words, F(z) is biharmonic on D if and only if F is harmonic on D. It is known [1] that a mapping F is biharmonic on D if and only if F can be represented as follow: F(z) = |z|2 G(z) + H (z), z ∈ D,
(1.1)
where G(z) and H (z) are complex-valued harmonic mappings on D. In [15], it’s known that a harmonic mapping f (z) is locally univalent on D if and only if its Jacobian J f (z) = | f z |2 − | f z |2 = 0 for any z ∈ D. Since D is simply connected, f (z) can be written as f = h + g with f (0) = h(0), h and g are analytic on D. Thus, we have J f (z) = |h (z)|2 − |g (z)|2 . For such function f , we define f (z) = max |eiθ f z (z) + e−iθ f z (z)| = | f z (z)| + | f z (z)|, 0≤θ≤2π
and λ f (z) = min |eiθ f z (z) + e−iθ f z (z)| = || f z (z)| − | f z (z)||. 0≤θ≤2π
Recall that a mapping ω : D → is said to be L 1 -Lipschitz (L 1 > 0) (l1 -coLipschitz (l1 > 0)) if |ω(z 1 ) − ω(z 2 )| ≤ L 1 |z 1 − z 2 |, z 1 , z 2 ∈ D, (|ω(z 1 ) − ω(z 2 )| ≥ l1 |z 1 − z 2 |, z 1 , z 2 ∈ D).
(1.2) (1.3)
A mapping ω is bi-Lipschitz if it is Lipschitz and co-Lipschitz (see [14]). In [13], the Lipschitz character of q.c. harmonic self-mappings of the unit disk was established with respect to the hyperbolic metric and this was generalized to an arbitrary domain in [25]. Harmonic mappings techniques have been u
Data Loading...
