Integral means and coefficient estimates on planar harmonic mappings

Planar Harmonic Univalent and Related Mappings

The theory of harmonic univalent mappings has become a very popular research topic in recent years. The aim of this expository article is to present a guided tour of the planar harmonic univalent and related mappings with emphasis on recent results and open problems and, in particular, to look at the harmonic analogues of the theory of analytic univalent functions in the unit disc.

Coefficient inequality for perturbed harmonic mappings

Landau's theorem for planar harmonic mappings

Landau gave a lower estimate for the radius of a schlicht disk centered at the origin and contained in the image of the unit disk under a bounded holomorphic function f normalized by f(0) = f ′(0)− 1 = 1. Chen, Gauthier, and Hengartner established analogous versions for bounded harmonic functions. We improve upon their estimates.

Convolutions of Planar Harmonic Convex Mappings

Ruscheweyh and Sheil-Small proved that convexity is preserved under the convolution of univalent analytic mappings in K. However, when we consider the convolution of univalent harmonic convex mappings in K H , this property does not hold. In fact, such convolutions may not be univalent. We establish some results concerning the convolution of univalent harmonic convex mappings provided that it i...

Inradius and Integral Means for Green’s Functions and Conformal Mappings

Let D be a convex planar domain of finite inradius RD . Fix the point 0 ∈ D and suppose the disk centered at 0 and radius RD is contained in D. Under these assumptions we prove that the symmetric decreasing rearrangement in θ of the Green’s function GD(0, ρe iθ), for fixed ρ, is dominated by the corresponding quantity for the strip of width 2RD . From this, sharp integral mean inequalities for ...