Schwarz Lemma Research Articles

Near-isometric duality of Hardy norms with applications to harmonic mappings

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.

Schwarz-type lemma at the boundary for mappings satisfying non-homogeneous polyharmonic equations

In this paper, we establish the boundary Schwarz lemma for solutions to non-homogeneous polyharmonic equations defined on the unit disk.

A Schwarz lemma at the boundary on complex Hilbert balls and applications to starlike mappings

In this paper, we prove a Schwarz lemma at the boundary for holomorphic mappings f between Hilbert balls, and obtain related consequences. Especially, we obtain estimations of ∥Df(z0)∥ on the holomorphic tangent space for holomorphic mappings f or for homogeneous polynomial mappings f between Hilbert balls. Next, we prove the boundary rigidity theorem for holomorphic self-mappings of a Hilbert ball which have an interior fixed point. We obtain two distortion theorems for various subclasses of starlike mappings on the Euclidean unit ball $${\mathbb{B}^n}$$ in ℂn, as applications of the boundary Schwarz lemma for holomorphic mappings between the Euclidean unit balls. Finally, certain coefficient bounds for subclasses of starlike mappings on the unit ball of a complex Hilbert space are also obtained as an application of the estimation of ∥Df(z0)∥ on the holomorphic tangent space for homogeneous polynomial mappings f between Hilbert balls.

Schwarz lemma for harmonic mappings into a geodesic line in a Riemann surfaces

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.

Schwarz-Type Lemmas Associated to a Helmholtz Equation

As a case of the Laplacian, the Helmholtz operator can be factorized using perturbed Dirac operators. In this article, we prove Schwarz lemmas for solutions of perturbed Dirac operators in vector space $$\mathbb {R}^{3}$$ by the integral representation formulas. As an immediate consequence of the Schwarz lemma, one has Liouville’s theorem for monogenic functions in Clifford analysis.

Boundary Schwarz lemma and rigidity property for holomorphic mappings of the unit polydisc in Cn

In this paper, we generalize the classical Schwarz lemma at the boundary from the unit disk D in the complex plane to the unit polydisc Dn in higher-dimensional complex space. Two boundary Schwarz lemmas for holomorphic mappings of Dn and corresponding rigidity properties are established without the restriction of the interior fixed point.

Hyperbolic metric on the strip and the Schwarz lemma for HQR mappings

We give simple proofs of various versions of the Schwarz lemma for real valued harmonic functions and for holomorphic (more generally harmonic quasiregular, shortly HQR) mappings with the strip codomain. Along the way, we get a simple proof of a new version of the Schwarz lemma for real valued harmonic functions (without the assumption that 0 is mapped to 0 by the corresponding map). Using the Schwarz-Pick lemma related to distortion for harmonic functions and the elementary properties of the hyperbolic geometry of the strip we get optimal estimates for modulus of HQR mappings.

A note on the Schwarz lemma for harmonic functions

In this note we consider some generalizations of the Schwarz lemma for harmonic functions on the unit disk, whereby values of such functions and the norms of their differentials at the point z = 0 are given.

A boundary Schwarz lemma for pluriharmonic mappings between the unit polydiscs of any dimensions

In this paper, we present a boundary Schwarz lemma for pluriharmonic mappings between the unit polydiscs of any dimensions, which extends the classical Schwarz lemma for bounded harmonic functions to higher dimensions.

Sharpened forms of analytic functions concerned with Hankel determinant

In this paper, we present a Schwarz lemma at the boundary for analytic functions at the unit disc, which generalizes classical Schwarz lemma for bounded analytic functions. For new inequalities, the results of Jack's lemma and Hankel determinant were used. We will get a sharp upper bound for Hankel determinant $H_{2}(1)$. Also, in a class of analytic functions on the unit disc, assuming the existence of angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained.

A form of Schwarz's lemma and a bound for the Kobayashi metric on convex domains

We present a form of Schwarz's lemma for holomorphic maps between convex domains D1 and D2. This result provides a lower bound on the distance between the images of relatively compact subsets of D1 and the boundary of D2. This is a natural improvement of an old estimate by Bernal-González that takes into account the geometry of ∂D1. Using similar techniques, we also provide a new estimate for the Kobayashi metric on bounded convex domains.

Schwarz lemma at the boundary on the classical domain of type ℐ𝒱

Schwarz lemma at the boundary on the classical domain of type ℐ𝒱

BOUNDARY SCHWARZ LEMMA FOR SOLUTIONS TO NONHOMOGENEOUS BIHARMONIC EQUATIONS

We establish a boundary Schwarz lemma for solutions to nonhomogeneous biharmonic equations.

Results Related to the Chern–Yamabe Flow

Let $$(X,\omega _0)$$ be a compact complex manifold of complex dimension n endowed with a Hermitian metric $$\omega _0$$ . The Chern–Yamabe problem is to find a conformal metric of $$\omega _0$$ such that its Chern scalar curvature is constant. As a generalization of the Chern–Yamabe problem, we study the problem of prescribing Chern scalar curvature. We then estimate the first nonzero eigenvalue of Hodge–de Rham Laplacian of $$(X,\omega _0)$$ . On the other hand, we prove a version of conformal Schwarz lemma on $$(X,\omega _0)$$ . All these are achieved by using geometric flows related to the Chern–Yamabe flow. Finally, we prove the backwards uniqueness of the Chern–Yamabe flow.

A Schwarz Lemma for Harmonic Functions in the Real Unit Ball

We establish a precise Schwarz lemma for real-valued and bounded harmonic functions in the real unit ball of dimension n. This extends Chen’s Schwarz-Pick lemma for real-valued and bounded planar harmonic mapping.

Some Remarks on Positive Real Functions and Their Circuit Applications

In this paper, a boundary version of the Schwarz lemma has been considered for driving point impedance functions at s=0 point of the imaginary axis. Accordingly, under Z(0)=0condition, the modulus of the derivative of the Z(s) function has been considered from below. Here, Z(alfa), c1 and c2 coefficients of the Taylor expansion of the Z(s)=beta+c1(s-alfa)+... function have been used in the obtained inequalities. The sharpness of these inequalities has also been proved.

Schwarz Lemmas via the Pluricomplex Green’s Function

We prove a version of the Schwarz lemma for holomorphic mappings from the unit disk into the symmetric product of a Riemann surface. Our proof is function-theoretic and self-contained. The main novelty in our proof is the use of the pluricomplex Green's function. We also prove several other Schwarz lemmas using this function.

The Schwarz Lemma at the Boundary of the Non-Convex Complex Ellipsoids

Let B2,p:= {z ∈ ℂ2: ∣z1∣2+ ∣z2∣p < 1} (0 < p< 1). Then, B2,p(0 < p < 1) is a non-convex complex ellipsoid in ℂ2 without smooth boundary. In this article, we establish a boundary Schwarz lemma at z0 ∈ ∂B2,p for holomorphic self-mappings of the non-convex complex ellipsoid B2, p, where z0 is any smooth boundary point of B2,p.

Schwarz lemma for driving point impedance functions and its circuit applications

SummaryIn this paper, a boundary version of the Schwarz lemma is investigated for driving point impedance functions and its circuit applications. It is known that driving point impedance function, Z(s) = 1 + cp(s − 1)p + cp + 1(s − 1)p + 1 + ..., p &gt; 1, is an analytic function defined on the right half of the s‐plane. Two theorems are presented using the modulus of the derivative of driving point impedance function, |Z′(0)|, by assuming the Z(s) function is also analytic at the boundary point s = 0 on the imaginary axis with . In the obtained inequalities, the value of the function at s = 1 and the derivatives with different orders have been used. Finally, the sharpness of the inequalities obtained in the presented theorems is proved. Simple LC circuits are obtained using the obtained driving point impedance functions.

The 3-Point Spectral Pick Interpolation Problem and an Application to Holomorphic Correspondences

We provide a necessary condition for the existence of a 3-point holomorphic interpolant $F:\mathbb{D}\longrightarrow\Omega_n$, $n\geq 2$. Our condition is inequivalent to the necessary conditions hitherto known for this problem. The condition generically involves a single inequality and is reminiscent of the Schwarz lemma. We combine some of the ideas and techniques used in our result on the $\mathcal{O}(\mathbb{D},\Omega_n)$-interpolation problem to establish a Schwarz lemma -- which may be of independent interest -- for holomorphic correspondences from $\mathbb{D}$ to a general planar domain $\Omega\Subset \mathbb{C}$.