Improved Bohr’s inequality for locally univalent harmonic mappings

We prove several improved versions of Bohr’s inequality for the harmonic mappings of the form $f=h+\overline{g}$, where $h$ is bounded by 1 and $\left|g^{\prime }\left(z\right)\right|\le \left|h^{\prime }\left(z\right)\right|$. The improvements are obtained along the lines of an earlier work of Kayumov and Ponnusamy, i.e. (Kayumov and Ponnusamy, 2018) for example a term related to the area of the image of the disk $D(0,r)$ under the mapping $f$ is considered. Our results are sharp. In addition, further improvements of the main results for certain special classes of harmonic mappings are provided.     

On lengths,areas and Lipschitz continuity of polyharmonic mappings

In this paper, we continue our investigation of polyharmonic mappings in the complex plane. First, we establish two Landau type theorems. We also show a three circles type theorem and an area version of the Schwarz lemma. Finally, we study Lipschitz continuity of polyharmonic mappings with respect to the distance ratio metric.     

Characterizations of Lipschitz-type,Bloch-type and Dirichlet-type spaces

Monatshefte für Mathematik - In this paper, we will give new characterizations of Lipschitz-type spaces, and establish related growth theorems for Bloch-type spaces. Then we will present...     

Conjugate function method for numerical conformal mappings

We present a method for numerical computation of conformal mappings from simply or doubly connected domains onto so-called canonical domains, which in our case are rectangles or annuli. The method is based on conjugate harmonic functions and properties of quadrilaterals. Several numerical examples are given.     

Landau’s theorem for polyharmonic mappings

In this paper, we first investigate coefficient estimates for bounded polyharmonic mappings in the unit disk D$\mathbb{D}$. Then, we obtain two versions of Landau’s theorem for polyharmonic mappings F$F$, and for the mappings of the type L(F)$L\left(F\right)$, where L$L$ is the differential operator of Abdulhadi, Abu Muhanna and Khuri. Examples and numerical estimates are given.     

Radial growth,Lipschitz and Dirichlet spaces on solutions to the non-homogenous Yukawa equation

In this paper, we investigate some properties of solutions f to the nonhomogenous Yukawa equation Δf(z) = λ(z)f(z) in the unit ball \(\mathbb{B}^n\) of ? ⁿ , where λ is a real function from \(\mathbb{B}^n\) into ?. First, we prove that a main result of Girela, Pavlovi? and Peláez (J. Analyse Math. 100 (2006), 53–81) on analytic functions can be extended to this more general setting. Then we study relationships on such solutions between the bounded mean oscillation and Lipschitz-type spaces. The obtained result generalized the corresponding result of Dyakonov (Acta Math. 178 (1997), 143–167). Finally, we discuss Dirichlet-type energy integrals on such solutions in the unit ball of ? ⁿ and give an application.     

Linear Connectivity,Schwarz–Pick Lemma and Univalency Criteria for Planar Harmonic Mapping

In this paper, we first establish a Schwarz–Pick lemma for higher-order derivatives of planar harmonic mappings, and apply it to obtain univalency criteria. Then we discuss distortion theorems,Lipschitz continuity and univalency of planar harmonic mappings defined in the unit disk with linearly connected images.     

Free distance ratio mappings in Banach spaces

Monatshefte für Mathematik - In this paper, we introduce a class of mappings related to the distance ratio metric and study its connection to the freely quasiconformal mapping in Banach...