Univalence criterion for harmonic mappings and Φ-like functions

In this paper, we introduce a new class of harmonic univalent functions with respect to k-symmetric points by using a newly-defined q-analog of the derivative operator for complex harmonic functions. For this harmonic univalent function class, we derive a sufficient condition, a representation theorem, and a distortion theorem. We also apply a generalized q-Bernardi–Libera–Livingston integral operator to examine the closure properties and coefficient bounds. Furthermore, we highlight some known consequences of our main results. In the concluding part of the article, we have finally reiterated the well-demonstrated fact that the results presented in this article can easily be rewritten as the so-called (p,q)-variations by making some straightforward simplifications, and it will be an inconsequential exercise, simply because the additional parameter p is obviously unnecessary.

By applying Ruscheweyh derivative on the class ASH(λ, α, k, γ )o f harmonic univalent functions in the unit disk U , we obtain several interesting properties such as sharp coefficient relations, distortion bounds, extreme points, Hadamard product, and other results.

The aim of this paper is twofold. First, we investigate the properties of the composition of harmonic mappings with harmonic mappings, and the composition of biharmonic mappings with harmonic mappings. Second, we consider the Goodman-Saff conjecture for biharmonic mappings in the unit disk. In fact, we show that the answer to the Goodman-Saff conjecture is positive for a special class of univalently biharmonic mappings which contains the set of all harmonic univalent mappings.

We shall start with the analytic foundations and, building upon this, treat existence, regularity, and uniqueness of harmonic maps; in the course of that, already some geometric applications will occur. The A role of harmonic maps in Riemannian and Kahlerian geometry will then be discussed somewhat more systematically in the subsequent paragraphs. The last paragraph finally is devoted to a novel development of Teichmuller theory via harmonic maps.

Harmonic pre-Schwarzian and its applications

We consider the class of all sense-preserving complex-valued harmonic mappings f = h + g ¯ f=h+\overline {g} defined on the unit disk D \mathbb {D} with the normalization h ( 0 ) = h ′ ( 0 ) − 1 = 0 h(0)=h’(0)-1=0 and g ( 0 ) = g ′ ( 0 ) = 0 g(0)=g’(0)=0 with the second complex dilatation ω : D → D \omega :\,\mathbb {D}\rightarrow \mathbb {D} , g ′ ( z ) = ω ( z ) h ′ ( z ) g’(z)=\omega (z)h’(z) . In this paper, the authors determine sufficient conditions on h h and ω \omega that would imply the univalence of harmonic mappings f = h + g ¯ f=h+\overline {g} on D \mathbb {D} .

<abstract><p>In this paper, for the first time, we apply symmetric $ q $ -calculus operator theory to define symmetric Salagean $ q $-differential operator. We introduce a new class $ \widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) $ of harmonic univalent functions $ f $ associated with newly defined symmetric Salagean $ q $-differential operator for complex harmonic functions. A sufficient coefficient condition for the functions $ f $ to be sense preserving and univalent and in the same class is obtained. It is proved that this coefficient condition is necessary for the functions in its subclass $ \overline{\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) } $ and obtain sharp coefficient bounds, distortion theorems and covering results. Furthermore, we also highlight some known consequence of our main results.</p></abstract>

We define and investigate a special class of Salagean-type harmonic univalent functions in the open unit disk. We obtain coefficient conditions, extreme points, distortion bounds, convex combinations for the above class of harmonic univalent functions. 2000 Mathematics Subject Classification. Primary 30C45 ; Secondary 30C50

The Einstein/Maxwell equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities phi: R^3\Sigma -> H^2_C, where Sigma is a subset of the axis of symmetry, and H^2_C is the complex hyperbolic plane. Motivated by this problem, we prove the existence and uniqueness of harmonic maps with prescribed singularities phi: R^n\Sigma -> H, where Sigma is a submanifold of R^n of co-dimension at least 2, and H is a classical Riemannian globally symmetric space of noncompact type and rank one. This result, when applied to the black hole problem, yields solutions which can be interpreted as equilibrium configurations of multiple co-axially rotating charged black holes held apart by singular struts.

We prove the existence and uniqueness of harmonic maps in degree one homotopy classes of closed, orientable surfaces of positive genus, where the target has non-positive Gauss curvature and conic points with cone angles less than 2π. For a homeomorphism w of such a surface, we prove existence and uniqueness of minimizers in the homotopy class of w relative to the inverse images of the cone points with cone angles less than or equal to π. The latter can be thought of as minimizing maps from punctured Riemann surfaces into conic surfaces. We discuss the regularity of these maps near the inverse images of the cone points in detail. For relative minimizers, we relate the gradient of the energy functional with the Hopf differential.When the genus is zero, we prove the same relative minimization provided there are at least three cone points of cone angle less than or equal to π.

Let ϕ(z) be a fixed harmonic functions of the form $\varphi (z) = z + \sum\nolimits_{k = 2}^\infty {c_k z^k + } \overline {\sum\nolimits_{k = 1}^\infty {d_k z^k } } $ (d k ≥ c k ≥ c 2 > 0; k ≥ 2) and S H (c k , d k , δ) be the subclass of harmonic univalent functions f(z) which satisfy the inequality $\sum\nolimits_{k = 2}^\infty {c_k |a_k | + } \sum\nolimits_{k = 1}^\infty {d_k |b_k | \leqslant \delta } $ . In this paper, we establish some interesting results on the ratio of harmonic univalent functions to its sequences of partial sums. Relevant connections of the results presented here with various known results are briefly indicated.

Analytic and Harmonic Univalent Functions

Functions f in the class $ K_H $ are convex, univalent, harmonic, and sense preserving in the unit disk. Such functions can be expressed as $ f = h + \\overline {g} $ where h and g are analytic functions. If $ f \\in K_H $ has $ h(0) = 0, g(0) = 0, h'(0) = 1$ , and $ g'(0) = 0 $ , then $ f \\in K_H^0 $ . For $ f \\in K_H^0 $ and } analytic in the unit disk, an integral representation for $ f\ ilde {*}\\varphi = h*\\varphi + \\overline {g*\\varphi } $ is found. With } a strip mapping, $ f\ ilde {*}\\varphi $ is shown to be in $ K_H^0 $ . In a 1958 paper, Pólya and Schoenberg conjectured that if f and g are conformal mappings of the unit disk onto convex domains, then the Hadamard product f 2 g of f and g has the same property. It is known that the analogue of that result for harmonic mappings is false. In this paper, some examples are given in which the property of convexity is preserved for Hadamard products of certain convex harmonic mappings. In addition, an integral formula is used to determine the geometry of the Hadamard product from the geometry of the factors. This is true in particular for the convolution of strip mappings with certain functions $ f_n \\in K_H^0 $ which take the unit disk to regular n -gons.

The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 ⩽ ρ < 1, the author constructs a closed convex domain E r,ρ such that $$F\left( {\bar \Delta \left( {z,r} \right)} \right) \subset e^{i\alpha } E_{r,\rho } = \left\{ {e^{i\alpha } z:z \in E_{r,\rho } } \right\}$$ holds for every z ∈ D, w = ρe iα and harmonic mapping F with F(D) ⊂ D and F(z) = w, where Δ(z, r) is the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρe iα and w′ ∈ e iα E r,ρ , there exists a harmonic mapping F such that F(D) ⊂ D, F(z) = w and F(z′) = w′ for some z′ ∈ ∂Δ(z, r). (II) The author establishes a Finsler metric H z (u) on the unit disk D such that $$\mathcal{H}_{F\left( z \right)} \left( {e^{i\theta } F_z \left( z \right) + e^{ - i\theta } F_{\bar z} \left( z \right)} \right) \leqslant \frac{1} {{1 - \left| z \right|^2 }}$$ holds for any z ∈ D, 0 ⩽ θ ⩽ 2π and harmonic mapping F with F(D) ⊂ D; furthermore, this result is precise and the equality may be attained for any values of z, θ, F(z) and arg\(\left( {e^{i\theta } Fz\left( z \right) + e^{ - i\theta } F_{\bar z} \left( z \right)} \right)\).

In this paper, a new subclass of complex-valued harmonic univalent functions f(z)=h(z)+g(z)¯¯¯¯¯¯¯¯¯ in the open disk U={z:z∈C and |z|&lt;1} defined by using a linear operator. We investigate coefficient condition, extreme points, distortion and convex combination for this subclass.