Related papers: Radius of injectivety for harmonic mappings with f…
We investigate harmonic mappings $f=h+\bar{g}$ defined in the unit disk, where $g$ and $h$ satisfy certain prescribed conditions and the analytic part $h$ belongs to the Ma and Minda class of starlike functions. Certain sharp radius results…
For a univalent smooth mapping $f$ of the unit disk $\ID$ of complex plane onto the manifold $f(\ID)$, let $d_f(z_0)$ be the radius of the largest univalent disk on the manifold $f(\ID)$ centered at $f(z_0)$ ($|z_0|<1$). The main aim of the…
We obtain the sharp upper and lower bounds for the growth and distortion of the analytic parts $h$ of orientation-preserving harmonic mappings $f=h+\overline g$ (normalized in the standard way) that map the unit disk onto a convex domain.
In this article, we determine the radius of univalence of sections of normalized univalent harmonic mappings for which the range is convex (resp. starlike, close-to-convex, convex in one direction). Our result on the radius of univalence of…
Let $f=h+\overline{g}$ be a normalized harmonic mapping in the unit disk $\ID$. In this paper, we obtain the sharp radius of univalence, fully starlikeness and fully convexity of the harmonic linear differential operators…
In this paper, we investigate properties of harmonic entire mappings. Firstly, we give the characterizations of the order and the type for a harmonic entire mapping $f=h+\overline{g}$, respectively, and also consider the relationship…
In the article the authors consider the class ${\mathcal H}_0$ of sense-preserving harmonic functions $f=h+\overline{g}$ defined in the unit disk $|z|<1$ and normalized so that $h(0)=0=h'(0)-1$ and $g(0)=0=g'(0)$, where $h$ and $g$ are…
In this article, we determine two point distortion theorem and sharp coefficient estimates for the families of close-to-convex harmonic mappings whose analytic part is a convex function of order $\alpha$. By making use of these results, we…
In this paper, we introduces and undertake as a systematical investigation of the class $\mathcal{P}_{\mathcal{H}}^{0}(\alpha,M)$ of normalized harmonic mappings $f = h + \overline{g}$ in the unit disk $\mathbb{D}$, defined by the…
The present article concerns the Bohr radius for $K$-quasiconformal sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ for which the analytic part $h$ is subordinated to some analytic function $\varphi$, and…
We determine the cut locus of arbitrary non-simply connected, compact and irreducible Riemannian symmetric space explicitly, and compute injectivity radius and diameter for every type of them.
We consider the class of all sense-preserving harmonic mappings $f= h+\overline{g}$ of the unit disk $\ID$, where $h$ and $g$ are analytic with $g(0)=0$, and determine the Bohr radius if any one of the following conditions holds: \bee $h$…
Let $\ast$ and $\widetilde {\ast}$ denote the convolution of two analytic maps and that of an analytic map and a harmonic map respectively. Pokhrel [1] proved that if $f = h+\overline{g}$ is a harmonic map convex in the direction of…
Let $\mathcal{S}_H^0$ denote the class of all functions $f(z)=h(z)+\overline{g(z)}=z+\sum^\infty_{n=2} a_nz^n +\overline{\sum^\infty_{n=2} b_nz^n}$ that are sense-preserving, harmonic and univalent in the open unit disk $|z|<1$. The…
We construct, for every \(0<k<1\), a bounded globally univalent harmonic mapping \[ f=h+\overline g \colon \D\to\C \] such that \[ |g'(z)|\le k|h'(z)|,\qquad z\in\D, \] while the analytic part \(h\) is unbounded. The construction is based…
The main aim of this paper is to obtain the sharp upper and lower bounds for the growth and distortion of the analytic part $h$ of sense-preserving convex $K$-quasiconformal harmonic mappings.
The hereditary property of convexity and starlikeness for conformal mappings does not generalize to univalent harmonic mappings. This failure leads us to the notion of fully starlike and convex mappings of order \alpha, (0\leq \alpha<1). A…
In 1984, Clunie and Sheil-Small proved that a sense-preserving harmonic function whose analytic part is convex, is univalent and close-to-convex. In this paper, certain cases are discussed under which the conclusion of this result can be…
In a 2D conservative Hamiltonian system there is a formal integral $\Phi$ besides the energy H. This is not convergent near a stable periodic orbit, but it is convergent near an unstable periodic orbit. We explain this difference and we…
In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold $M$: 1) the convexity radius of $p$,…