Schwarz Pick Type Research Articles

Schwarz-Pick type inequalities from an operator theoretical point of view

We use (versions of) the von Neumann inequality for Hilbert space contractions to prove several Schwarz-Pick type inequalities. Specifically, we derive an alternate proof for a multi-point Schwarz-Pick inequality by Beardon and Minda, along with a generalized version for operators. Connections with model spaces and Peschl's invariant derivatives are established. Finally, Schwarz-Pick inequalities for analytic functions on polydisks and for higher order derivatives are discussed. An enhanced version of the Schwarz-Pick lemma, using the notion of distinguished variety, is obtained for the bidisk.

The p-Bohr radius for vector-valued holomorphic and pluriharmonic functions

Abstract We study a “p-powered” version K n p ⁢ ( F ⁢ ( R ) ) {K_{n}^{p}(F(R))} of the well-known Bohr radius problem for the family F ⁢ ( R ) {F(R)} of holomorphic functions f : R → X {f:R\to X} satisfying ∥ f ∥ < ∞ {\lVert f\rVert<\infty} , where ∥ ⋅ ∥ {\lVert\,\cdot\,\rVert} is a norm in the function space F ⁢ ( R ) {F(R)} , R ⊂ ℂ n {R\subset{\mathbb{C}}^{n}} is a complete Reinhardt domain, and X is a complex Banach space. For all p > 0 {p>0} , we describe in full detail the asymptotic behavior of K n p ⁢ ( F ⁢ ( R ) ) {K_{n}^{p}(F(R))} , where F ⁢ ( R ) {F(R)} is: (a) the Hardy space of X-valued holomorphic functions defined in the open unit polydisk 𝔻 n {{\mathbb{D}}^{n}} ; and (b) the space of bounded X-valued holomorphic or complex-valued pluriharmonic functions defined in the open unit ball B ⁢ ( l t n ) {B(l_{t}^{n})} of the Minkowski space l t n {l_{t}^{n}} . We give an alternative definition of the optimal cotype for a complex Banach space X in the light of these results. In addition, the best possible versions of two theorems from [C. Bénéteau, A. Dahlner and D. Khavinson, Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4 2004, 1, 1–19] and [S. Chen and H. Hamada, Some sharp Schwarz–Pick type estimates and their applications of harmonic and pluriharmonic functions, J. Funct. Anal. 282 2022, 1, Paper No. 109254] have been obtained as specific instances of our results.

On Schwarz–Pick-Type Inequality and Lipschitz Continuity for Solutions to Nonhomogeneous Biharmonic Equations

The purpose of this paper is to study the Schwarz–Pick type inequality and the Lipschitz continuity for the solutions to the nonhomogeneous biharmonic equation: $$\Delta (\Delta f)=g$$ , where g : $$\overline{{\mathbb D}}\rightarrow {\mathbb {C}}$$ is a continuous function and $$\overline{{\mathbb D}}$$ denotes the closure of the unit disk $${\mathbb D}$$ in the complex plane $${\mathbb {C}}$$ . In fact, we establish the following properties for these solutions: First, we show that the solutions f do not always satisfy the Schwarz–Pick-type inequality $$\begin{aligned} \frac{1-|z|^2}{1-|f(z)|^2} \le C, \end{aligned}$$ where C is a constant. Second, we establish a general Schwarz–Pick-type inequality of f under certain conditions. Third, we discuss the Lipschitz continuity of f, and as applications, we get the Lipschitz continuity with respect to the distance ratio metric and the Lipschitz continuity with respect to the hyperbolic metric.

The Schwarz lemma in bicomplex analysis

We investigate Schwarz lemma in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero‐divisors. The bicomplex is a generalization of complex which has close relations with Fractal geometry, Minkowski Space‐Time, Maxwell's equations, Schrödinger equation, and Gaussian pulse wave. In this paper, we first construct a type of bicomplex Möbius transformation and obtain some results: the mapping properties on bicomplex sphere and bicomplex ball, preserving the inverse points with respect to the bicomplex sphere . Then we obtain the Poisson integral formula in bicomplex setting, and by using the Poisson integral formula, we give the Schwarz lemma for bicomplex holomorphic functions in bicomplex setting. Finally, we shall give the Schwarz lemma and the Schwarz–Pick type lemma for holomorphic functions in bicomplex analysis. These results may give new energy for the development of quantum mechanics.

Some generalizations for the Schwarz-Pick lemma and boundary Schwarz lemma

<abstract><p>In this paper, we first obtain a Schwarz-Pick type lemma for the holomorphic self-mapping of the unit disk with respect to the $ q $-distance. Second, we establish the general Schwarz-Pick lemma for the self-mapping of the unit disk satisfying the Poisson differential inequality. As an application, it is proven that this mapping is Lipschitz continuous with respect to the $ q $-distance under certain conditions. Moreover, the corresponding explicit Lipschitz constant is given. Third, it is proved that there exists a self-mapping of the unit disk satisfying the Poisson differential inequality, which does not meet conditions of the boundary Schwarz lemma. Finally, with some additional conditions, a boundary Schwarz lemma for the self-mapping of the unit disk satisfying the Poisson differential inequality is established.</p></abstract>

Schwarz–Pick Lemma for Harmonic and Hyperbolic Harmonic Functions

We establish some inequalities of Schwarz–Pick type for harmonic and hyperbolic harmonic functions on the unit ball of $${\mathbb {R}^n}$$ and we disprove a recent conjecture of Liu (Int Math Res Not, 2021).

Some sharp Schwarz-Pick type estimates and their applications of harmonic and pluriharmonic functions

The purpose of this paper is to study the Schwarz-Pick type inequalities for harmonic or pluriharmonic functions. By analogy with the generalized Khavinson conjecture, we first give some sharp estimates of the norm of harmonic functions from the Euclidean unit ball in Rn into the unit ball of the real Minkowski space. Next, we give several sharp Schwarz-Pick type inequalities for pluriharmonic functions from the Euclidean unit ball in Cn or from the unit polydisc in Cn into the unit ball of the Minkowski space. Furthermore, we establish some sharp coefficient type Schwarz-Pick inequalities for pluriharmonic functions defined in the Minkowski space. Finally, we use the obtained Schwarz-Pick type inequalities to discuss the Lipschitz continuity, the Schwarz-Pick type lemmas of arbitrary order and the Bohr phenomenon of harmonic or pluriharmonic functions.

Schwarz–Pick Lemma for Harmonic Functions

Abstract Based on the recently proven Khavinson conjecture, we establish an inequality of Schwarz–Pick type for harmonic functions on the unit ball of $\mathbb{R}^n$.

Schwarz–Pick and Landau Type Theorems for Solutions to the Dirichlet–Neumann Problem in the Unit Disk

The aim of this paper is to establish some properties of solutions to the Dirichlet–Neumann problem: $$(\partial _z\partial _{\overline{z}})^2 w=g$$ in the unit disc $${{\mathbb {D}}}$$ , $$w=\gamma _0$$ and $$\partial _{\nu }\partial _z\partial _{\overline{z}}w=\gamma $$ on $${\mathbb {T}}$$ (the unit circle), $$\frac{1}{2\pi i}\int _{{\mathbb {T}}}w_{\zeta {\overline{\zeta }}}(\zeta )\frac{d\zeta }{\zeta }=c$$ , where $$\partial _\nu $$ denotes differentiation in the outward normal direction. More precisely, we obtain Schwarz–Pick type inequalities and Landau type theorem for solutions to the Dirichlet–Neumann problem.

On Schwarz-Pick Type Inequality for Mappings Satisfying Poisson Differential Inequality

Let f be a twice continuously differentiable self-mapping of a unit disk satisfying Poisson differential inequality ∣Δf(z)∣ ≤ B · ∣Df (z)∣2 for some B > 0 and f(0) = 0. In this note, we show that f does not always satisfy the Schwarz-Pick type inequality $$\frac{1-\vert z\vert^{2}}{1-\vert f(z)\vert^{2}}\leq\ C(B),$$ where C(B) is a constant depending only on B. Moreover, a more general Schwarz-Pick type inequality for mapping that satisfies general Poisson differential inequality is established under certain conditions.

On properties of solutions to the α-harmonic equation

The aim of this paper is to establish properties of solutions to the α-harmonic equations: , where is a continuous function and denotes the closure of the unit disc in the complex plane . We obtain Schwarz type and Schwarz-Pick type inequalities for solutions to the α-harmonic equation. In particular, for , solutions to the above equation are called α-harmonic functions. We determine the necessary and sufficient conditions for an analytic function ψ to have the property that is α-harmonic function for any α-harmonic function f. Furthermore, we discuss the Bergman-type spaces on α-harmonic functions.

Schwarz-Pick Type Estimates for Gradients of Pluriharmonic Mappings of the Unit Ball

The main purpose of this paper is to find generalized Schwarz-Pick type inequalities for pluriharmonic functions f of the unit ball $${\mathbb {B}}^n$$ . The estimates for the gradient of |f| and f are given as follows: $$\begin{aligned} \big |\nabla |f|(z)\big |\le \frac{4\sqrt{n}}{\pi (1-|z|)}, \ \ \ z\in {\mathbb {B}}^n \end{aligned}$$ and $$\begin{aligned} |\nabla f(z)|^2\le \frac{2n(1-|f(z)|^2)}{1-|z|}, \ \ \ z\in {\mathbb {B}}^n. \end{aligned}$$ Moreover, by additional assuming the quasiregularity of f, some additional estimates are given in this paper.

Euclidean maximum moduli of plane domains and their applications

ABSTRACTFor a plane domain we study correlations of the Euclidean maximum modulus and three hyperbolic domain characteristics connected with the Poincaré metric of the domain and the distance function. We prove that the Laplacian of the hyperbolic radius of every domain of hyperbolic type is a subharmonic function. Also, for any doubly connected domain we prove asymptotically sharp estimates for the hyperbolic characteristics using the Euclidean maximum modulus of the domain. In addition, we obtain applications of these estimates to Hardy and Schwarz–Pick type inequalities.

The Schwarz Type Lemma in Upper Half Space in Clifford Analysis

In this paper, we firstly construct the Euler’s expression of hyper-complex numbers in $$H_{n+1}$$ , as an application of the Euler’s expression, we prove that the well-known homogeneous monogenic polynomials $$V_{l_1\dots l_p}(\mathbf{x})$$ are functions with values in $$H_{n+1}$$ . Secondly we construct a type of general Mobius transformation in Clifford analysis, the mapping properties are given; The Jacobi determinant and the monogenic property under these Mobius transformations are shown. Finally, by using the above Mobius transformation and modifying the Schwarz lemma in Zhang (J Math Anal Appl 443:1130–1141, 2016), we establish the Schwarz–Pick type lemmas for monogenic functions in the upper half space $$H^+_{n+1}$$ . A new generalization of the Schwarz lemma for harmonic functions with values in $$H_{n+1}$$ is also given.

Schwarz Pick type inequalities for harmonic maps between Riemann surfaces

ABSTRACTWe study the Schwarz lemma for harmonic mappings in general setting and prove among the other results that: if is the class of harmonic mappings f of the unit disk into the geodesic disk of a Riemann surface , which is disjoint from its cut-locus with radius M satisfying some curvature restrictions, then there is a locally Lipschitz homeomorphism , convex in , with and such that for with we have for.

SCHWARZ LEMMA FOR HOLOMORPHIC MAPPINGS IN THE UNIT BALL

AbstractIn this note, we establish a Schwarz–Pick type inequality for holomorphic mappings between unit balls Bn and Bm in corresponding complex spaces. We also prove a Schwarz-Pick type inequality for pluri-harmonic functions.

Several properties of $$\alpha $$ α -harmonic functions in the unit disk

The aim of this paper is to obtain the Schwarz–Pick type inequality for $$\alpha $$ -harmonic functions f in the unit disk and get estimates on the coefficients of f. As an application, a Landau type theorem of $$\alpha $$ -harmonic functions is established.

Inequalities of Schwarz–Pick type with branch points and the pseudo-hyperbolic distance

The classical Schwarz–Pick lemma implies that a holomorphic function of the open unit disk into itself gives a contraction with respect to the hyperbolic metric on the disk. As such, differential forms of the Schwarz–Pick lemma have been recently discussed in consideration of branch points of a holomorphic function. We in this article demonstrate that the pseudo-hyperbolic distance, which does not arise from a Riemannian metric, remains suitable for a Schwarz–Pick type of inequality even in more general settings with the existence of branch points.

Equality statements for Schwarz–Pick type estimates and the pseudo-hyperbolic distance

In the present article we demonstrate that the conclusion on the case of equality for the estimate of Schwarz–Pick type is a property of a distance-nonincreasing holomorphic mapping which is locally defined. Also, while considering existence of branch points, we deal with the topics in terms of the pseudo-hyperbolic distance.

The Schwarz lemma for functions with values in C(Vn,0)

In this paper, we first construct a type of Möbius transformations with Clifford coefficients and give some of its properties, for example: the mapping properties, preserving the symmetric points with respect to the sphere, the Jacobi determinant, the monogenic properties under these Möbius transformations. Then by using the integral representation formulas for harmonic functions with values in a Clifford algebra C(Vn,0) and the integral inequality estimation, we give the Schwarz lemma for harmonic functions with values in a Clifford algebra C(Vn,0), which is sharper than the known results. Combining the Schwarz lemma with the Möbius transformations, the Schwarz–Pick type lemmas for functions with values in C(Vn,0) are given. These results greatly improve the results in [21].