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In this paper, we first obtain a refined version of the Bohr inequality of norm-type for holomorphic mappings with lacunary series on the polydisk in $ \mathbb {C}^n $ C n under some restricted conditions. Next, we determine the refined version of the Bohr inequality for holomorphic functions defined on a balanced domain G of a complex Banach space X and take values in the unit disk $ \mathbb {D} $ D . Furthermore, as a consequence of one of these results, we obtain a refined version of the Bohr-type inequality for harmonic functions $ f=h+\bar {g} $ f = h + g ¯ defined on a balanced domain $ G\subset X $ G ⊂ X. All the results are proved to be sharp. [ABSTRACT FROM AUTHOR]

In this paper, we first derive a Landau-type theorem and a Bloch-type theorem for the class of meromorphic harmonic mappings. Then we establish two new versions of Bloch-type theorems for certain K-quasiregular meromorphic harmonic mappings in the unit disk. [ABSTRACT FROM AUTHOR]

Let \Sigma _H^k(p) be the class of sense-preserving univalent harmonic mappings defined on the open unit disk \mathbb {D} of the complex plane with a simple pole at z=p \in (0,1) that have k-quasiconformal extensions (0\leq k<1) onto the extended complex plane. In this article, we obtain an area theorem for this class of functions. [ABSTRACT FROM AUTHOR]

Let α > - 1 and assume that f is α -harmonic mapping defined in the unit disk that belongs to the Hardy class h p with p ⩾ 1 . We obtain some sharp estimates of the type | f (z) | ≤ g (| r |) ‖ f ∗ ‖ p and | D f (z) | ≤ h (| r |) ‖ f ∗ ‖ p . We also prove a Schwarz type lemma for the class of α -harmonic mappings of the unit disk onto itself fixing the origin. [ABSTRACT FROM AUTHOR]

Let A = { z : r < | z | < R } and A ∗ = { z : r ∗ < | z | < R ∗ } be annuli in the complex plane. Let p ∈ [ 1 , 2 ] and assume that H 1 , p (A , A ∗) is the class of Sobolev homeomorphisms between A and A ∗ , h : A → onto\,\, A ∗ . Then we consider the following Dirichlet type energy of h: F p [ h ] = ∫ A ‖ D h ‖ p | h | p , 1 ⩽ p ⩽ 2. We prove that this energy integral attains its minimum, and the minimum is a certain radial diffeomorphism h : A → onto\,\, A ∗ , provided a radial diffeomorphic minimizer exists. If p > 1 then such diffeomorphism exists always. If p = 1 , then the conformal modulus of A ∗ must not be greater or equal to π / 2 . This curious phenomenon is opposite to the Nitsche type phenomenon known for the standard Dirichlet energy. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate properties of certain class of logharmonic mappings. Initially, we establish the argument principle of sense-preserving mappings F = h g ¯ + H G ¯ , where h, g, H and G are analytic functions. As applications, we obtain a direct extension of Rouché’s theorem, open mapping theorem and minimal area image of sense-preserving logharmonic mappings. Furthermore, we present an estimate for the modulus of the partial derivative of K-quasiconformal logharmonic mappings, and by using it, we get a Schwarz type lemma of K-quasiconformal logharmonic mappings. Finally, we obtain estimates of h ′ ′ (0) 2 and g ′ ′ (0) 2 for univalent sense-preserving logharmonic mappings f = h g ¯ in the unit disk. [ABSTRACT FROM AUTHOR]

In this paper, we first establish two versions of Landau–Bloch type theorem for (K , K ′) -elliptic harmonic mappings with a bounded minimum distortion. Next, we provide several coefficient estimates and a conjecture for (K , K ′) -elliptic harmonic mappings. Then, we establish three new versions of Landau–Bloch type theorem for sense-preserving harmonic mappings. Finally, we establish a sharp version of Landau–Bloch type theorem for certain harmonic mappings. These results are sharp in some given cases and improve the related results of different authors. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the Riesz theorem for harmonic quasiregular mappings for a special case (when p = 2) in the unit disc. Our results improve similar results in this field and are proved with milder conditions. Moreover, we prove another variant forms of Riesz inequality for harmonic quasiregular functions. [ABSTRACT FROM AUTHOR]

Let I be a line segment in the complex plane C . We describe a method of constructing a bi-Lipschitz sense-preserving mapping of C onto itself, which is harmonic in C \ I and coincides with a given sufficiently regular function f : I → C . As a result we show that a quasiconformal self-mapping of C which is harmonic in C \ I does not have to be harmonic in C . [ABSTRACT FROM AUTHOR]

A survey of works concerning difficulties associated with harmonic grid generation in plane domains with angles and cutouts is given, and some new results are presented. It is well known that harmonic grids produced by standard methods in domains with cutouts or reentrant angles (i.e., interior angles greater than π) may contain defects, such as self-overlappings or exit beyond the domain boundary. It is established that, near the vertex of a reentrant angle, these defects follow from the asymptotics constructed for the underlying harmonic mapping, according to which the grid line leaving the angle vertex is tangent to one of the angle sides at the vertex (an effect referred to as "adhesion"), except for a special case. A survey of results is given for domains of three types with angles or cutouts (L-shaped, horseshoe, and a domain with a rectangular cutout), for which standard methods for harmonic grid generation encounter difficulties. Applying the multipole method to such domains yields a harmonic mapping for them with high accuracy: the a posteriori error estimate of the mapping in the norm is 10–7 in the case of using 120 approximative functions. [ABSTRACT FROM AUTHOR]

This paper aims to introduce a measure of the non-univalency of a harmonic mapping. By using it, we find the best approximation of an analytic function by a univalent harmonic mapping. [ABSTRACT FROM AUTHOR]

In this article, we prove a criterion for quasiconformal extension of the integral transforms of analytic functions defined in the unit disk of the complex plane. Thereafter, we prove a sufficient condition for quasiconformal extensibility of sense-preserving harmonic mappings defined in the unit disk and applying this result, we obtain criteria for the quasiconformal extension of various types of integral transforms that are defined for sense-preserving harmonic mappings. We also study the stability of the quasiconformal extension of sense-preserving harmonic mappings. [ABSTRACT FROM AUTHOR]

We investigate harmonic mappings f = h + 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 univalency, close-to-convexity, fully starlikeness, fully convexity, and radius of strongly starlikeness are established. We also calculate the radius of uniformly starlikeness and convexity for these functions. Several results enhance the well-known radius results. [ABSTRACT FROM AUTHOR]

We establish two-point distortion theorems for sense-preserving planar harmonic mappings f = h + g ¯ in the unit disk D which satisfy harmonic versions of the univalence criteria due to Becker and Nehari. In addition, we also find two-point distortion theorems for the cases when h is a normalized convex function and, more generally, when h (D) is a c-linearly connected domain. [ABSTRACT FROM AUTHOR]

Assume that f is a real ρ -harmonic function of the unit disk $\mathbb{D}$ onto the interval $(-1,1)$ , where $\rho(u,v)=R(u)$ is a metric defined in the infinite strip $(-1,1)\times \mathbb{R}$. Then we prove that $|\nabla f(z)|(1-|z|^2)\le \frac{4}{\pi}(1-f(z)^2)$ for all $z\in\mathbb{D}$ , provided that ρ has a non-negative Gaussian curvature. This extends several results in the field and answers to a conjecture proposed by the first author in 2014. Such an inequality is not true for negatively curved metrics. [ABSTRACT FROM AUTHOR]

A function (meromorphic or harmonic) from the hyperbolic disk of the complex plane to the Riemann sphere is normal if its dilatation is bounded. In this paper, normal families of meromorphic functions is studied in view of sharing values by their differential monomials. Moreover, several properties of normal harmonic mappings are also studied in view of certain general settings. [ABSTRACT FROM AUTHOR]

By operating with the Scale Relativity Theory by means of two scenarios (Schrӧdinger and Madelung-type scenarios) in the dynamics of complex systems, we can achieve a description of these complex systems through a holographic-type perspective. Then, a gauge invariance of the Riccati type becomes functional in complex-system dynamics, which implies several consequences: conservation laws (in particular, for dynamics, the kinetic momentum conservation law), simultaneity and synchronization among the structural units' (belonging to a complex system) dynamics, and temporal patterns through harmonic mappings. Finally, an economic case analysis is highlighted. [ABSTRACT FROM AUTHOR]

We study the mappings of the Orlicz–Sobolev classes with a branching defined in the unit ball of the Euclidean space. We have obtained estimates for the distortion of the distance under these mappings at the points of the unit sphere. Under some conditions, we have also obtained the Hölder continuity of the mappings mentioned above. If we suppose that the considered mappings are solutions to certain Laplacian-gradient inequalities, we get the Lipschitz property. In section 7–9, we review some results, prove a new result, Theorem 7.1, and outline the proof of Theorem 9.2. [ABSTRACT FROM AUTHOR]

We consider the class of stable harmonic mappings f = h + g ¯ introduced by Martin, Hernandez and the class of stable logharmonic mappings f = z h g ¯ introduced by AbdulHadi, El-Hajj. We determine Bohr’s radius for the classes of stable univalent harmonic mappings, stable convex harmonic mappings and stable univalent logharmonic mappings. We also consider improved and refined versions of Bohr’s inequality and discuss the Bohr–Rogosinski’s radius for this family of mappings. [ABSTRACT FROM AUTHOR]

We prove Schwarz-type lemma results for Weierstrass parameterization of the minimal disk in the Riemannian manifold M × R , where M is a Riemannian surface of non-positive Gaussian curvature. The estimate is sharp, and the equality is attained if and only if the ϱ -harmonic mapping that produces the parameterization is conformal and the metric is a Euclidean metric. If the area of the minimal surface is equal to the area of the disk, then the parametrization is a contraction w.r.t. induced metric and hyperbolic metric respectively. [ABSTRACT FROM AUTHOR]

We prove the following. If f is a harmonic quasiconformal mapping between the unit ball in ℝ n and a spatial domain with C1,α boundary, then f is Lipschitz continuous in B. This generalizes some known results for n = 2 and improves some others in higher dimensional case. [ABSTRACT FROM AUTHOR]

In this paper, we establish two new versions of Landau-type theorems for pluriharmonic mappings with a bounded distortion. Then using these results, we derive three Bloch-type theorems of pluriharmonic mappings, which improve the corresponding results of Chen and Gauthier. [ABSTRACT FROM AUTHOR]

By operating with the Scale Relativity Theory by means of two scenarios (Schrӧdinger and Madelung-type scenarios) in the dynamics of complex systems, we can achieve a description of these complex systems through a holographic-type perspective. Then, a gauge invariance of the Riccati type becomes functional in complex-system dynamics, which implies several consequences: conservation laws (in particular, for dynamics, the kinetic momentum conservation law), simultaneity and synchronization among the structural units’ (belonging to a complex system) dynamics, and temporal patterns through harmonic mappings. Finally, an economic case analysis is highlighted.

In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions of completely monotone sequences. [ABSTRACT FROM AUTHOR]

Let F = H ÷ G be a locally injective and sense-preserving harmonic mapping of the unit disk D in the complex plane C, where H and G arc holomorphic in D and G(O) = 0. The aim of this paper is studying interplay between properties of Fe := H + eG, e - C, and its holomorphic part H. In particular, several results dealing with the injectivity of Fe arc obtained. [ABSTRACT FROM AUTHOR]

We prove the following result. If f is a harmonic quasiconformal mapping between two Jordan domains D and Ω having C¹ boundaries, then the function f is globally Hölder continuous for every α<1 but it is not necessarily Lipschitz in general. This result extends and improves a classical theorem of S. Warschawski for conformal mappings. [ABSTRACT FROM AUTHOR]

It is well-known that two locally univalent analytic functions have equal Schwarzian derivative if and only if each one of them is a composition of the other with a non-constant Möbius transformation. The main goal in this paper is to extend this result to the cases when the functions considered are harmonic. That is, we identify completely the transformations that preserve the (harmonic) Schwarzian derivative of locally univalent harmonic functions. [ABSTRACT FROM AUTHOR]

In this paper we obtain the sharp Bohr radius for a family of bounded analytic functions B` and for the family of sensepreserving K-quasiconformal harmonic mappings of the form f = h + ¬g , where h ∈ B`

Abstract Let ψ μ , ν ( z ) = ( 1 − 2 cos ν e i μ z + e 2 i μ z 2 ) − 1 $\psi_{\mu,\nu}(z)=(1-2\cos\nu e^{i\mu}z+e^{2i\mu}z^{2})^{-1}$ , μ , ν ∈ [ 0 , 2 π ) $\mu,\nu\in[0,2\pi)$ and p be an analytic mapping with Re p > 0 $\operatorname{Re} p>0$ on the open unit disk. We consider the sense-preserving planar harmonic mappings f = h + g ‾ $f=h+\overline{g}$ , which are shears of the mapping ∫ 0 z ψ μ , ν ( ξ ) p ( ξ ) d ξ $\int_{0}^{z} \psi_{\mu,\nu}(\xi) p(\xi)\,{d}\xi$ in the direction μ. These mappings include the harmonic right half-plan mappings, vertical strip mappings, and their rotations. For various choices of dilatations g ′ / h ′ $g'/h'$ of f, sufficient conditions are found for the convex combinations of these mappings to be univalent and convex in the direction μ.

The prime purpose of this article aims to establish connections between various sub-classes of harmonic univalent mappings by applying a convolution operator associated with the well-known Wright function. In particular, the inclusion relations among Goodman–Rønning-type harmonic univalent mappings, harmonic k-uniformly convex mappings and harmonic k-uniformly starlike mappings in the plane are established. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of sense preserving harmonic mappings f = h + g ¯ in the open unit disk and prove that functions in this class are close-to-convex. We give some basic properties such as coefficient bounds, growth estimates, convolution and determine the radius of convexity for the sections of functions belonging to this family. In addition, we construct certain harmonic univalent polynomials belonging to this family. [ABSTRACT FROM AUTHOR]

A harmonic mapping is a univalent harmonic function of one complex variable. We define a family of harmonic mappings on the unit disk whose images are rotationally symmetric “rosettes” with n cusps or n nodes, n ≥ 3 . These mappings are analogous to the n-cusped hypocycloid, but are modified by Gauss hypergeometric factors, both in the analytic and co-analytic parts. Relative rotations by an angle β of the analytic and anti-analytic parts lead to graphs that have cyclic, and in some cases dihedral symmetry of order n. While the graphs for different β can be dissimilar, the cusps are aligned along axes that are independent of β . For certain isolated values of β , the boundary function is continuous with arcs of constancy, and has nodes of interior angle π / 2 - π / n instead of cusps. [ABSTRACT FROM AUTHOR]

The plane domain D is called R-convex if D contains each compact set bounded by two shortest sub-arcs of the radius R with endpoints w1, w2 ∈ D, |w1−w2|

Suppose w is a sense-preserving harmonic mapping of the unit disk D such that w (D) ⊆ D and w has a zero of order p ≥ 1 at z = 0 . In this paper, we first improve the Schwarz lemma for w, and then, we establish its boundary Schwarz lemma. Moreover, by using the automorphism of D , we further generalize this result. [ABSTRACT FROM AUTHOR]

Let W be the wedge { z : arg z ∈ (- π / 4 , π / 4) } in the complex plane C and assume that D is the unit disk. Assume further that Σ ⊂ C × R is a minimal surface over W. We obtain some sharp point-wise estimates of Gram–Schmidt norm of the first derivative of a harmonic diffeomorpism of the unit disk onto W. Further we apply this result to obtain some upper estimates of the Gaussian curvature of the minimal surface Σ at a given point Q in term of underlaying point w ∈ W . This improves some result by Abu-Muhanna and Schober (Can J Math 39(6):1489–1530, 1987). [ABSTRACT FROM AUTHOR]

We prove a Schwarz type lemma for harmonic mappings between the unit disk and a geodesic line in a Riemann surface. It presents a certain extension of the Schwarz lemma for the real harmonic mappings with respect to the Euclidean metric defined on the unit disk. [ABSTRACT FROM AUTHOR]

We consider the problem of minimizing the weighted Dirichlet energy between homeomorphisms of planar annuli. A known challenge lies in the case when the weight λ depends on the independent variable z. We prove that for an increasing radial weight λ(z) the infimal energy within the class of all Sobolev homeomorphisms is the same as in the class of radially symmetric maps. For a general radial weight λ(z) we establish the same result in the case when the target is conformally thin compared to the domain. Fixing the admissible homeomorphisms on the outer boundary we establish the radial symmetry for every such weight. [ABSTRACT FROM AUTHOR]

We study harmonic mappings of the form f (z) = h (z) − z ¯ , where h is an analytic function. In particular, we are interested in the index (a generalized multiplicity) of the zeros of such functions. Outside the critical set of f, where the Jacobian of f is non-vanishing, it is known that this index has similar properties as the classical multiplicity of zeros of analytic functions. Little is known about the index of zeros on the critical set, where the Jacobian vanishes; such zeros are called singular zeros. Our main result is a characterization of the index of singular zeros, which enables one to determine the index directly from the power series of h. [ABSTRACT FROM AUTHOR]

We find a simple expression in complex terms for homogeneous harmonic polynomials, which we use to express the Laurent series of a harmonic function around an isolated singularity. Also, we show a residue theorem and study the orientation at isolated singularities through the use of complex dilatation, focusing on those points where orientation is not preserved nor reversed, making essential the concept of exceptional set and extending it to isolated singularities. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let denote the class of all complex-valued harmonic functions f in the open unit disk normalized by f(0) = fz(0) − 1 = = 0, and the subclasses of consisting of univalent and sense-preserving functions and normalized analytic functions, respectively. For φ ∈ , let := {f = h + ḡ ∈ : h – e2αig = φ} be subfamily of. In this paper, we shall determine the conditions under which the analytic function φ with φ ∈ , the linear convex combination tf1 + (1 − t)f2 with fj ∈ , j = 1, 2, and the harmonic convolution f1 ∗ f2 with fj ∈ , j = 1, 2, are univalent and convex in one direction, respectively. Many previous related results are generalized. [ABSTRACT FROM AUTHOR]