Let $f = P[F]$ denote the Poisson integral of $F$ in the unit disk $\mathbb{D}$ with $F$ is an absolute continuous in the unit circle $\mathbb{T}$ and $\dot{F}\in L^p(\mathbb{T})$, where $\dot{F}(e^{it}) = \frac{d}{dt} F(e^{it})$ and $p \in [1,\infty]$. Recently, Chen et al. (J. Geom. Anal., 2021) extended Zhu's results (J. Geom. Anal., 2020) and proved that (i) if $f$ is a harmonic mapping and $1 \leq p < \infty$, then $f_z$ and $\overline{f_{\overline{z}}} \in B^p(\mathbb{D})$, the Bergman spaces of $\mathbb{D}$. Moreover, (ii) under additional conditions as $f$ being harmonic quasiregular mapping in \cite{Zhu} or $f$ being harmonic elliptic mapping in \cite{CPW}, they proved that $f_z$ and $\overline{f_{\overline{z}}}\in H^p(\mathbb{D})$, the Hardy space of $\mathbb{D}$, for $1 \leq p \leq \infty$. The aim of this paper is to extend these results by showing that (ii) holds for $p\in(1,\infty)$ without any extra conditions and for $p=1$ or $p=\infty$, $f_z$ and $\overline{f_{\bar{z}}}\in H^p(\mathbb{D})$ if and only if $H(\dot{F})\in L^p(\mathbb{T})$, the Hilbert transform of $\dot{F}$ and in that case, it yields $zf_z=P[\frac{\dot{F}+iH(\dot{F})}{2i}]$.
We review the relation between the classical formulas of the pre-Schwarzian and Schwarzian derivatives of locally univalent analytic functions and the derivatives of the generating functions of the methods due to Newton and Halley, respectively, for approximating zeros. We extend these relations to the cases when the functions considered are harmonic.
In this paper we introduce a definition of the pre-Schwarzian and the Schwarzian derivatives of any locally univalent harmonic mapping $f$ in the complex plane without assuming any additional condition on the (second complex) dilatation $ω_f$ of $f$. Using the new definition for the Schwarzian derivative of harmonic mappings, we prove analogous theorems to those by Chuaqui, Duren, and Osgood. Also, we obtain a Becker-type criterion for the univalence of harmonic mappings.
In this paper, we extend Ahlfors's univalent criteria and Ahlfors's quasiconformal extension for analytic functions to harmonic mappings defined in the unit disk. Moreover, we give a general quasiconformal extension of harmonic Teichmüller mappings, whose maximal dilatation estimate is asymptotically sharp.
We prove that if the Schwarzian norm of a given complex-valued locally univalent harmonic mapping $f$ in the unit disk is small enough, then $f$ is, indeed, globally univalent and can be extended to a quasiconformal mapping in the extended complex plane.
Suppose $p\geq1$, $w=P[F]$ is a harmonic mapping of the unit disk $\mathbb{D}$ satisfying $F$ is absolutely continuous and $\dot{F}\in L^p(0, 2π)$, where $\dot{F}(e^{it})=\frac{\mathrm{d}}{\mathrm{d}t}F(e^{it})$. In this paper, we obtain Bergman norm estimates of the partial derivatives for $w$, i.e., $\|w_z\|_{L^p}$ and $\|\overline{w_{\bar{z}}}\|_{L^p}$, where $1\leq p<2$. Furthermore, if $w$ is a harmonic quasiregular mapping of $\mathbb{D}$, then we show that $w_z$ and $\overline{w_{\bar{z}}}$ are in the Hardy space $H^p$, where $1\leq p\leq\infty$. The corresponding Hardy norm estimates, …
It is well-known that two locally univalent analytic functions $φ$ and $ψ$ have equal Schwarzian derivative if and only if there exists a non-constant Möbius transformation $T$ such that $φ=T\circ ψ$. In this paper, we identify completely the relationship between two locally univalent harmonic mappings with equal (harmonic) Schwarzian derivative.
Suppose $α>-1$ and $1\leq p \leq \infty$. Let $f=P_α[F]$ be an $α$-harmonic mapping on $\mathbb{D}$ with the boundary $F$ being absolute continuous and $\dot{F}\in L^p(0,2π)$, where $\dot{F}(e^{iθ}):=\frac{dF(e^{iθ})}{dθ}$. In this paper, we investigate the membership of $f_z$ and $f_{\overline{z}}$ in the space $\mathcal{H}_{\mathcal{G}}^{p}(\mathbb{D})$, the generalized Hardy space. We prove, if $α>0$, then both $f_z$ and $f_{\overline{z}}$ are in $\mathcal{H}_{\mathcal{G}}^{p}(\mathbb{D})$. If $α<0$, then $f_z$ and $f_{\overline{z}}\in \mathcal{H}_{\mathcal{G}}^{p}(\mathbb{D})$ if and only …
For analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass--Enneper lifts of planar harmonic mappings to their associated minimal surfaces. Finally certain classes of harmonic mappings are shown to have finite Schwarzian norm.
In this paper we have introduced two new classes $\mathcal{H}\mathcal{M}(β, λ, k, ν)$ and $\overline{\mathcal{H}\mathcal{M}} (β, λ, k, ν)$ of complex valued harmonic multivalent functions of the form $f = h + \overline g$, satisfying the condition \[ Re \{(1 - λ) \frac{Ω^vf}{z} + λ(1-k) \frac{(Ω^vf)'}{z'} + λk \frac{(Ω^vf)''}{z''} \} > β, (z\in \mathcal{D})\] where $h$ and $g$ are analytic in the unit disk $\mathcal{D} = \{z : |z| < 1\}.$ A sufficient coefficient condition for this function in the class $\mathcal{H}\mathcal{M}(β, λ, k, ν)$ and a necessary and sufficient coefficient condition f…
Let $f = P[F]$ denote the Poisson integral of $F$ in the unit disk $\mathbb{D}$ with $F$ being absolutely continuous in the unit circle $\mathbb{T}$ and $\dot{F}\in L_p(0, 2π)$, where $\dot{F}(e^{it})=\frac{d}{dt} F(e^{it})$ and $p\geq 1$. Recently, the author in \cite{Zhu} proved that $(1)$ if $f$ is a harmonic mapping and $1\leq p< 2$, then $f_{z}$ and $\overline{f_{\overline{z}}}\in \mathcal{B}^{p}(\mathbb{D}),$ the classical Bergman spaces of $\mathbb{D}$ \cite[Theorem 1.2]{Zhu}; $(2)$ if $f$ is a harmonic quasiregular mapping and $1\leq p\leq \infty$, then $f_{z},$ $\overline{f_{\overline…
Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to be uniformly locally univalent with respect to the hyperbolic metric if and only if it has finite Schwarzian norm, thus generalizing a result of B. Schwarz for analytic functions. A numerical bound is obtained for the Schwarzian norms of univalent harmonic mappings.
We give the lower bound for the modulus of the radial derivatives and Jacobian of harmonic injective mappings from the unit ball onto convex domain in plane and space. As an application we show co-Lipschitz property of some classes of qch mappings. We also review related results in planar case using some novelty.
The primary aim of this article is to extend certain inequalities concerning the pre-Schwarzian derivatives from the case of analytic univalent functions to that of univalent harmonic mappings defined on certain domains. This is done in two different ways. One of the ways is to connect with a conjecture on the univalent harmonic mappings. Also, we improve certain known results on the majorization of the Jacobian of functions in the affine and linear invariant family of sense-preserving harmonic mappings. This is achieved as an application of a corresponding distortion theorem in terms of the h…
We provide two new formulas for quasiconformal extension to $\overline{\mathbb{C}}$ for harmonic mappings defined in the unit disk and having sufficiently small Schwarzian derivative. Both are generalizations of the Ahlfors-Weill extension for holomorphic functions.
We prove that a harmonic quasiconformal mapping defined on a finitely connected domain in the plane, all of whose boundary components are either points or quasicircles, admits a quasiconformal extension to the whole plane if its Schwarzian derivative is small. We also make the observation that a univalence criterion for harmonic mappings holds on uniform domains.
We study subharmonic functions whose Laplacian is supported on a null set and in connected components of of the complement to the support admit harmonic extensions to larger sets. We prove that if such a function has a piecewise holomorphic derivative, then it is locally piecewise harmonic. In generic cases it coincides locally with the maximum of finitely many harmonic functions. Moreover, we describe the support when the holomorphic derivative satisfies a global algebraic equation. The proofs follow classical patterns and our methods may also be of independent interest.
It is proved that on any smoothly bounded domain $D$ in $\mathbb{R}^n$, $n>1$, the output of the harmonic Bergman projection belongs to the Sobolev space of order $k$ whenever all tangential derivatives of order up to k of the input function belong to $L^2(D)$.
We determine the representation theorem, distortion theorem, coefficients estimate and Bohr's radius for log-harmonic starlike mappings of order $α$, which are generalization of some earlier results. In addition, the inner mapping radius of log-harmonic mappings is also established by constructing a family of $1$-slit log-harmonic mappings. Finally, we introduce pre-Schwarzian, Schwarzian derivatives and Bloch's norm for non-vanishing log-harmonic mappings, several properties related to these are also obtained.
The theory of $q$-analogs frequently occurs in a number of areas, including the fractals and dynamical systems. The $q$-derivatives and $q$-integrals play a prominent role in the study of $q$-deformed quantum mechanical simple harmonic oscillator. In this paper, we define a symmetric $q$-derivative operator and study new family of univalent functions defined by use of that operator. We establish some new relations between functions satisfying analytic conditions related to conical sections.
In earlier work the authors have extended Nehari's well-known Schwarzian derivative criterion for univalence of analytic functions to a univalence criterion for canonical lifts of harmonic mappings to minimal surfaces. The present paper develops some quantitative versions of that result in the form of two-point distortion theorems. Along the way some distortion theorems for curves in ${\Bbb R}^n$ are given, thereby recasting a recent injectivity criterion of Chuaqui and Gevirtz in quantitative form.
Hardy spaces in the complex plane and in higher dimensions have natural finite-dimensional subspaces formed by polynomials or by linear maps. We use the restriction of Hardy norms to such subspaces to describe the set of possible derivatives of harmonic self-maps of a ball, providing a version of the Schwarz lemma for harmonic maps. These restricted Hardy norms display unexpected near-isometric duality between the exponents 1 and 4, which we use to give an explicit form of harmonic Schwarz lemma.
In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative $S(f)$ of a locally univalent analytic function $f$ in the unit disk satisfies that $\limsup_{|z|\to 1} |S(f)(z)| (1-|z|^2)^2 < 2$, then there exists a positive integer $N$ such that $f$ takes every value at most $N$ times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function $f$ satisfies that $\limsup_{|z|\to 1} |f"(z)/f'(z)|\, (1-|z|^2)< 1$. In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when $f$ is merely harmonic…
It is proved the existence of nonclassical solutions of the Neumann problem for the harmonic functions in the Jordan rectifiable domains with arbitrary measurable boundary distributions of normal derivatives. The same is stated for the partial case of the Poincare problem on directional derivatives. Moreover, it is shown that the spaces of the found solutions have the infinite dimension.
In this paper, we investigate the Bohr radius for $K$-quasiregular sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ such that the translated analytic part $h(z)-h(0)$ is quasi-subordinate to some analytic function. The main aim of this article is to extend and to establish sharp versions of four recent theorems by Liu and Ponnusamy \cite{LP2019} and, in particular, we settle affirmatively the two conjectures proposed by them. Furthermore, we establish two refined versions of Bohr's inequalities and determine the Bohr radius for the derivatives of analytic fun…
This paper deals with an extremal problem for harmonic functions in the unit ball of $\mathbf{R}^n$. We are concerned with the pointwise sharp estimates for the gradient of real--valued bounded harmonic functions. Our main result may be formulated as follows. The sharp constants in the estimates for the absolute value of the radial derivative and the modulus of the gradient of a bounded harmonic function coincide near the boundary of the unit ball. This result partially confirms a conjecture posed by D. Khavinson.
A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass--Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian defined for conformal metrics and on a Schwarzian introduced by Ahlfors for curves. Convexity plays a central role.
We present a theory of ultradistributional boundary values for harmonic functions defined on the Euclidean unit ball. We also give a characterization of ultradifferentiable functions and ultradistributions on the sphere in terms of their spherical harmonic expansions. To this end, we obtain explicit estimates for partial derivatives of spherical harmonics, which are of independent interest and refine earlier estimates by Calderon and Zygmund. We apply our results to characterize the support of ultradistributions on the sphere via Abel summability of their spherical harmonic expansions.
Using the estimate of the difference between the discrete harmonic function and its corresponding continuous version we derive a rate of convergence of the Loewner driving function for the harmonic explorer to the Brownian motion with speed 4 on the real line. Based on this convergence rate, the derivative estimate for chordal $\mbox{SLE}_4$, and the estimate of tip structure modulus for harmonic explorer paths, we obtain an explicit power-law rate of convergence of the harmonic explorer paths to the trace of chordal $\mbox{SLE}_4$ in the supremum distance.
We study harmonic Besov spaces $b^p_α$ on the unit ball of $\mathbb{R}^n$, where $0<p<1$ and $α\in\mathbb{R}$. We provide characterizations in terms of partial and radial derivatives and certain radial differential operators that are more compatible with reproducing kernels of harmonic Bergman-Besov spaces. We show that the dual of harmonic Besov space $b^p_α$ is weighted Bloch space $b_β^{\infty}$ under certain volume integral pairing for $0<p<1$ and $α,β\in\mathbb{R}$. Our other results are about growth at the boundary and atomic decomposition.
We present some new relations between the Cauchy-Riemann operator on the real Clifford algebra $\mathbb R_n$ of signature $(0,n)$ and slice-regular functions on $\mathbb R_n$. The class of slice-regular functions, which comprises all polynomials with coefficients on one side, is the base of a recent function theory in several hypercomplex settings, including quaternions and Clifford algebras. In this paper we present formulas, relating the Cauchy-Riemann operator, the spherical Dirac operator, the differential operator characterizing slice regularity, and the {spherical derivative} of a slice …
We give a local characterization of the class of functions having positive distributional derivative with respect to $\bar{z}$ that are almost everywhere equal to one of finitely many analytic functions and satisfy some mild non-degeneracy assumptions. As a consequence, we give conditions that guarantee that any subharmonic piecewise harmonic function coincides locally with the maximum of finitely many harmonic functions and we describe the topology of their level curves. These results are valid in a quite general setting as they assume no {\em à priori} conditions on the differentiable struct…
For every $m\in\mathbb{N}$, we establish the equidistribution of the sequence of the averaged pull-backs of a Dirac measure at any given value in $\mathbb{C}\setminus\{0\}$ under the $m$-th order derivatives of the iterates of a polynomials $f\in \mathbb{C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of $f$ with pole at $\infty$. We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field $k$ for a sequence of effective divisors on $\mathbb{P}…
