Landau-Bloch theorems for bounded biharmonic mappings

We determine coefficient bounds for bounded planar biharmonic mappings and generalize the Landau-Bloch univalency theorems for such bounded biharmonic functions. The univalence radii presented here improve many related results published to date, including the most recent one [Complex Var. Elliptic Equ. 58(12) (2013), 1667-1676] and are sharp in some given cases.

Generalized Poisson integral and sharp estimates for harmonic and biharmonic functions in the half-space

A representation for the sharp coefficient in a pointwise estimate for the gradient of a generalized Poisson integral of a function f on ℝn−1 is obtained under the assumption that f belongs to Lp. It is assumed that the kernel of the integral depends on the parameters α and β. The explicit formulas for the sharp coefficients are found for the cases p = 1, p = 2 and for some values of α, β in the case p = ∞. Conditions ensuring the validity of some analogues of the Khavinson’s conjecture for the generalized Poisson integral are obtained. The sharp estimates are applied to harmonic and biharmonic functions in the half-space.

On Surface Completion and Image Inpainting by Biharmonic Functions: Numerical Aspects

Numerical experiments with smooth surface extension and image inpainting using harmonic and biharmonic functions are carried out. The boundary data used for constructing biharmonic functions are the values of the Laplacian and normal derivatives of the functions on the boundary. Finite difference schemes for solving these harmonic functions are discussed in detail.